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From a photograph by Mr. LEWIS M. RUTHERFORD. 



& Romance of tlje Jftfloon. 













tcmbers 0! tljc fonbait Institution 




HAVING been honoured once again with a 
request that I should lecture before the London 
Institution, I chose for my subject the Theory of 
Tidal Evolution. The kind reception which these 
lectures received has led to their publication in 
the present volume. I have taken the opportunity 
to supplement the lectures as actually delivered 
by the insertion of some additional matter. I am 
indebted to my friends Mr. Close and Mr. Ram- 
baut for their kindness in reading the proofs. 


Observatory, Co. DUBLIN, 
April 26, 1889. 


IN the present Edition I have taken the 
opportunity of making sundry verbal corrections. 
I may also remark that the reader is presumed 
to be acquainted with such ordinary astronomical 
facts as would be contained in a book of the 
scope of my little volume Starland. 


May ist, 1892. 



IT is my privilege to address you this afternoon 
on a subject in which science and poetry are 
blended in a happy conjunction. If there be a 
peculiar fascination about the earlier chapters of 
any branch of history, how great must be the 
interest which attaches to that most primeval of 
all terrestrial histories which relates to the actual 
beginnings of this globe on which we stand. 

In our efforts to grope into the dim recesses of 
this awful past, we want the aid of some steadfast 
light which shall illumine the dark places without 
the treachery of the will-o'-the-wisp. In the 
absence of that steadfast light, vague conjectures 
as to the beginning of things could never be 


entitled to any more respect than was due to 
mere matters of speculation. 

Of late, however, the required light has been to 
some considerable extent forthcoming, and the 
attempt has been made, with no little success, to 
elucidate a most interesting and wonderful chapter 
of an exceedingly remote history. To chronicle 
this history is the object of the present lectures 
before this Institution. 

First, let us be fully aware of the extraordinary 
remoteness of that period of which our history 
treats. To attempt to define that period chrono- 
logically would be utterly futile. When we have 
stated that it is more ancient than almost any other 
period which we can discuss, we have expressed 
all that we are really entitled to say. Yet this 
conveys not a little. It directs us to look back 
through all the ages of modern human history, 
through the great days of ancient Greece and 
Rome, back through the times when Egypt and 
Assyria were names of renown, through the days 
when Nineveh and Babylon were mighty and 
populous cities in the zenith of their glory. Back 
earlier still to those more ancient nations of which 


we know hardly anything, and still earlier to 
the prehistoric man, of whom we know less ; back, 
finally, to the days when man first trod on this 
planet, untold ages ago. Here is indeed a por- 
tentous retrospect from most points of view, but 
it is only the commencement of that which our 
subject suggests. 

For man is but the final product of the long 
anterior ages during which the development of life 
seems to have undergone an exceedingly gradual 
elevation. Our retrospect now takes its way along 
the vistas opened up by the geologists. We look 
through the protracted tertiary ages, when mighty 
animals, now generally extinct, roamed over the 
continents. Back still earlier through those won- 
drous secondary periods, where swamps or oceans 
often covered what is now dry land, and where 
mighty reptiles of uncouth forms stalked and 
crawled and swam through the old world and the 
new. Back still earlier through those vitally sig- 
nificant ages when the sunbeams were being 
garnered and laid aside for man's use in the great 
forests, which were afterwards preserved by being 
transformed into seams of coal. Back still earlier 


through endless thousands of years, when lustrous 
fishes abounded in the oceans ; back again to those 
periods characterized by the lower types of life ; 
and still earlier to that incredibly remote epoch 
when life itself began to dawn on our awakening 
globe. Even here the epoch of our present history 
can hardly be said to have been reached. We 
have to look through a long succession of ages 
still antecedent. The geologist, who has hitherto 
guided our view, cannot render us much further 
assistance ; but the physicist is at hand he teaches 
us that the warm globe on which life is beginning 
has passed in its previous stages through every 
phase of warmth, of fervour, of glowing heat, of 
incandescence, and of actual fusion ; and thus at 
last our retrospect reaches that particular period 
of our earth's past history which is specially 
illustrated by the modern doctrine of Time and 

The present is the clue to the past. It is the 
steady application of this principle which has led 
to such epoch-making labours as those by which 
Lyell investigated the origin of the earth's crust, 
Darwin the origin of species, Max M tiller the origin 


of language. In our present subject the course 
is plain. Study exactly what is going on at 
present, and then have the courage to apply con- 
sistently and rigorously what we have learned 
from the present to the interpretation of the 

Thus we begin with the ripple of the tide on the 
sea-beach which we see to-day. The ebb and the 
flow of the tide are the present manifestations of 
an agent which has been constantly at work. Let 
that present teach us what tides must have done 
in the course of ages now past. 

It has been known from the very earliest times 
that the moon and the tides were connected 
together connected, I say, for a great advance 
had to be made in human knowledge before it 
would have been possible to understand the true 
relation between the tides and the moon. Indeed, 
that relation is so far from being of an obvious 
character, that I think I have read of a race who 
felt some doubt as to whether the moon was the 
cause of the tides, or the tides the cause of the 
moon. I should, however, say that the moon is 
not the sole agent engaged in producing this 


periodic movement of our waters. The sun also 
arouses a tide, but the solar tide is so small in 
comparison with that produced by the moon, that 
for our present purpose we may leave it out of 
consideration. We must, however, refer to the 
solar tide at a later period of our discourses, for 
it will be found to have played a splendid part 
at the initial stage of the Earth-Moon History, 
while in the remote future it will again rise into 

It will be well to set forth a few preliminary 
figures which shall explain how it comes to pass 
that the efficiency of the sun as a tide-producing 
agent is so greatly inferior to that of the moon. 
Indeed, considering that the sun has a mass so 
stupendous, that it controls the entire planetary 
system, it seems strange that a body so insignificant 
as the moon can raise a bigger tide on the ocean 
than can the sun, of which the mass is 26,000,000 
times as great as that of our satellite ? 

This apparent paradox will disappear when we 
enunciate the law according to which the efficiency 
of a tide-producing agent is to be estimated. This 
law is somewhat different from the familiar form in 


which the law of gravitation is expressed. The 
gravitation between two distant masses is to be 
measured by multiplying these masses together, 
and dividing the product by the square of the dis- 
tance. The law for expressing the efficiency of a 
tide-producing agent varies not according to the 
inverse square, but according to the inverse cube 
of the distance. This difference in the expression 
of the law will suffice to account for the superiority 
of the moon as a tide-producer over the sun. The 
moon's distance on an average is about one 
386th part of that of the sun, and thus it is easy 
to show that so far as the mere attraction of gravi- 
tation is concerned, the efficiency of the sun's force 
on the earth is about one hundred and seventy-five 
times as great as the force with which the moon 
attracts the earth. That is of course calculated 
under the law of the inverse square. To determine 
the tidal efficiency we have to divide this by three 
hundred and eighty-six, and thus we see that the 
tidal efficiency of the sun is less than half that of 
the moon. 

When the solar tide and the lunar tide are acting 1 


in unison, they conspire to produce very high 


tides and very low tides, or, as we call them, spring 
tides. On the other hand, when the sun is so 
placed as to give us a low tide while the moon is 
producing a high tide, the net result that we 
actually experience is merely the excess of the lunar 
tide over the solar tide ; these are what we call neap 
tides. In fact, by very careful and long-continued 
observations of the rise and fall of the tides at a 
particular port, it becomes possible to determine 
with accuracy the relative ranges of spring tides 
and neap tides ; and as the spring tides are pro- 
duced by moon plus sun, while the neap tides 
are produced by moon minus sun, we obtain a 
means of actually weighing the relative masses of 
the sun and moon. This is one of the remarkable 
facts which can be deduced from the careful study 
of the tides. 

The demonstration of the law of the tide-pro- 
ducing force is of a mathematical character, and 


I do not intend in these lectures to enter into 
mathematical calculations. There is, however, a 
simple line of reasoning which, though it falls short 
of actual demonstration, may yet suffice to give a 
plausible reason for the law. 


The tides owe their origin to the circumstance 
that the tide-producing agent operates more power- 
fully on those parts of the tide-exhibiting body 
which are near to it, than on the more distant por- 
tions of the same. The nearer the two bodies are 
together, the larger will be the relative differences 
in the distances of its various parts from the tide- 
producing body ; and on this account the leverage, 
so to speak, of the action by which the tides are 
produced is increased. For instance, if the two 
bodies were brought within half their original dis- 
tance of each other, the relative size of each body, 
as viewed from the other, will be doubled ; and 
what we have called the leverage of the tide-pro- 
ducing ability will be increased twofold. The 
gravitation also between the two bodies is in- 
creased fourfold when the distance is halved, and 
consequently, the tide-producing ability is doubled 
for one reason, and increased fourfold again by 
another ; hence, the tides will be increased eight- 
fold when the distance is reduced to one half. 
Now, as eight is the cube of two, this illustration 
may be taken as a verification of the law, that 
the efficiency of a body as a tide-producer varies 


inversely as the cube of the distance between it 
and the body on which the tides are being raised. 

For simplicity we may make the assumption 
that the whole of the earth is submerged beneath 
the ocean, and that the moon revolves in the plane 
of the equator. We may also entirely neglect for 
the present the tides produced by the sun, and we 
shall also make the further assumption that fric- 
tion is absent. What friction is capable of doing 
we must, however, refer to later on. The moon 
will act on the ocean and deform it, so that there 
will be high tide along one meridian, and high 
tide also on the opposite meridian. This is indeed 
one of the paradoxes by which students are 
frequently puzzled when they begin to learn 
about the tides. That the moon should pull the 
water into a heap on one side seems plausible 
enough. High tide will be there produced ; and the 
student might naturally think that the water being 
drawn in this way into a heap on one side, there 
will of course be low tide on the opposite side of 
the earth. A natural assumption, perhaps, but 
nevertheless a very wrong one. There are at 
every moment two opposite parts of the earth in 


a condition of high water ; in fact, this will be 
obvious if we remember that every day, or, to speak 
a little more accurately, in every twenty-four hours 
and fifty-one minutes, we have on the average 
two high tides at each locality. Of course this 
could not be if the moon raised only one heap 
of high water, because, as the moon only appears 
to revolve around the earth once a day, or, more 
accurately, once in that same average period of 
twenty-four hours and fifty-one minutes, it would 
be impossible for us to have high tides succeeding 
each other as they do in periods a little longer 
than twelve hours, if only one heap were carried 
round the earth. 

The first question then is, as to how these two 
opposite heaps of water are placed in respect to 
the position of the moon. The most obvious ex- 
planation would seem to be, that the moon should 
pull the waters up into a heap directly underneath 
it, and that therefore there should be high water 
underneath the moon. As to the other side, the 
presence of a high tide there was, on this theory, 
to be accounted for by the fact that the moon 
pulled the earth away from the waters on the more 

B 2 



remote side, just as it pulled the waters away from 
the more remote earth on the side underneath the 
moon. It is, however, certainly not the case that 
the high tide is situated in the simple position 
that this law would indicate, and which we have 
represented in Fig. I, where the circular body is 
the earth, the ocean surrounding which is distorted 
by the action of the tides. 

We have here taken an oval to represent the 
shape into which the water is supposed to be 
forced or drawn by the tidal action of the tide- 
producing body. This may possibly be a correct 
representation of what would occur on an ideal 


globe entirely covered with a frictionless ocean. 
But as our earth is not covered entirely by water, 
and as the ocean is very far from being frictionless, 
the ideal tide is not the tide that we actually know ; 
nor is the ideal tide represented by this oval even 
an approximation to the actual tides to which 
our oceans are subject. Indeed, the oval does not 
represent the facts at all, and of this it is only 
necessary to adduce a single fact in demonstration. 
I take the fundamental issue so often debated, as 
to whether in the ocean vibrating with ideal tides 
the high water or the low water should be under 
the moon. Or to put the matter otherwise ; when 
we represent the displaced water by an oval, is 
the long axis of the oval to be turned to the 
moon, as generally supposed, or is it to be directed 
at right angles therefrom ? If the ideal tides were 
in any degree representative of the actual tides, so 
fundamental a question as this could be at once 
answered by an appeal to the facts of observation. 
Even if friction in some degree masked the phe- 
nomena, surely one would think that the state of 
the actual tides should still enable us to answer 
this question. 


But a study of the tides at different ports fails 
to realize this expectation. At some ports, no 
doubt, the tide is high when the moon is on the 
meridian. In that case, of course, the high water 
is under the moon, as apparently ought to be the 
case invariably, on a superficial view. But, on 
the other hand, there are ports where there is 
often low water when the moon is crossing the 
meridian. Yet other ports might be cited in 
which every intermediate phase could be observed. 
If the theory of the tides was to be the simple one 
so often described, then at every port noon should 
be the hour of high water on the day of the new 
moon or of the full moon, because then both tide- 
exciting bodies are on the meridian at the same 
time. Even if the friction retarded the great tidal 
wave uniformly, the high tide on the days of full 
or change should always occur at fixed hours ; but, 
unfortunately, there is no such delightful theory 
of the tides as this would imply. At Greenock 
no doubt there is high water at or about noon on 
the day of full or change ; and if it could be simi- 
larly said that on the day of full or change there 
was high water everywhere at local noon, then 


the equilibrium theory of the tides, as it is called, 
would be beautifully simple. But this is not the 
case. Even around our own coasts the discre- 
pancies are such as to utterly discredit the theory 
as offering any practical guide. At Aberdeen 
the high tide does not appear till an hour later 
than the doctrine would suggest. It is two hours 
late at London, three at Tynemouth, four at Tralee, 
five at Sligo, and six at Hull. This last port 
would be indeed the haven of refuge for those 
who believe that the low tide ought to be under 
the moon. At Hull this is no doubt the case ; and 
if at all other places the water behaved as it does 
at Hull, why then, of course, it would follow that 
the law of low water under the moon was generally 
true. But then this would not tally with the con- 
dition of affairs at the other places I have named ; 
and to complete the cycle I shall add a few more. 
At Bristol the high water does not get up until 
seven hours after the moon has passed the meridian, 
at Arklow the delay is eight hours, at Yarmouth 
it is nine, at the Needles it is ten hours, while 
lastly, the moon has nearly got back to the 
meridian again ere it has succeeded in dragging 


up the tide on which Liverpool's great commerce 
so largely depends. 

Nor does the result of studying the tides along 
other coasts beside our own decide more conclu- 
sively on the mooted point. Even ports in the vast 
ocean give a very uncertain response. Kerguelen 
Island and Santa Cruz might seem to prove that 
the high tide occurs under the moon, but unfortu- 
nately both Fiji and Ascension seem to present us 
with an equally satisfactory demonstration, that 
beneath the moon is the invariable home of low 

I do not mean to say that the study of the tides 
is in other respects such a confused subject as the 
facts I have stated would seem to indicate. It 
becomes rather puzzling, no doubt, when we com- 
pare the tides at one port with the tides elsewhere. 
The law and order are, then, by no means con- 
spicuous, they are often hardly discernible. But 
when we confine our attention to the tides at a 
single port, the problem becomes at once a very 
intelligible one. Indeed, the investigation of the 
tides is an easy subject, if we are contented with 
a reasonably approximate solution ; should, how- 


ever, it be necessary to discuss fully the tides at 
any port, the theory of the method necessary for 
doing so is available, and an interesting and 
beautiful theory it certainly is. 

Let us then speak for a few moments about the 
methods by which we can study the tides at a 
particular port. The principle on which it is based 
is a very simple one. 

It is the month of August, the i8th, we shall 
suppose, and we are going to enjoy a delicious 
swim in the sea. We desire, of course, to secure a 
high tide for the purpose of doing so, and we 
call an almanac to help us. I refer to Thorn's 
Dublin Directory, where I find the tide to be high 
at loh. I4m. on the morning of the iSth of August. 
That will then be the time to go down to the 
baths at Howth or Kingstown. 

But what I am now going to discourse to you 
about is not the delights of sea-bathing, it is rather 
a different inquiry. I want to ask, How did the 
people who prepared that almanac know years 
beforehand, that on that particular day the tide 
would be high at that particular hour? How do 
they predict for every day the hour of high water ? 


and how comes it to pass that these predictions are 
invariably correct ? 

We first refer to that wonderful book, the Nau- 
tical Almanac. In that volume the movements of 
the moon are set forth with full detail ; and among 
other particulars we can learn on page iv of every 
month the mean time of the moon's meridian 
passage. It appears that on the day in question 
the moon crossed the meridian at nh. 23m. Thus 
we see there was high water at Dublin at loh. 
I4m., and ih. Qm. later, that is, at uh. 23m., the 
moon crossed the meridian. 

Let us take another instance. There is a high 
tide at 3.40 P.M. on the 25th August, and again 
the infallible Nautical Almanac tells us that the 
moon crossed the meridian at 5h. 44m., that is, at 
2h. 4m. after the high water. 

In the first case the moon followed the tide in 
about an hour, and in the second case the moon 
followed in about two hours. Now if we are to be 
satisfied with a very rough tide rule for Dublin, 
we may say generally that there is always a high 
tide an hour and a half before the moon crosses 
the meridian. This would not be a very accurate 


rule, but I can assure you of this, that if you go 
by it you will never fail of finding a good tide to 
enable you to enjoy your swim. I do not say this 
rule would enable you to construct a respectable 
tide-table. A ship-owner who has to creep up the 
river, and to whom the inches of water are often 
material, will require far more accurate tables than 
this simple rule could give. But we enter into 
rather complicated matters when we attempt to 
give any really accurate methods of computation. 
On these we shall say a few words presently. 
What I first want to do, is to impress upon you in 
a simple way the fact of the relation between the 
tide and the moon. 

To give another illustration, let us see how the 
tides at London Bridge are related to the moon. 
On Jan. ist, 1887, it appeared that the tide was 
high at 6h. 26m. P.M., and that the moon had 
crossed the meridian 56m. previously ; on the 8th 
Jan. the tide was high at oh. 43m. P.M., and the 
moon had crossed the meridian 2h. im. previously. 
Therefore we would have at London Bridge high 
water following the moon's transit in somewhere 
about an hour and a half. 


I choose a day at random, for example the 
1 2th April. The moon crosses the upper meridian 
at 3h. 3901. A.M., and the lower meridian at 4-h 
6m. P.M. Adding an hour and a half to each 
would give the high tides at $h. 901. A.M. and 
5h. 36m. P.M. ; as a matter of fact, they are 4h. 
58m. A.M. and $h. 2om. P.M. 

But these illustrations are sufficient. We find 
that at London, in a general way, high water 
appears at London Bridge about an hour and a 
half after the moon has passed the meridian of 
London. It so happens that the interval at Dublin 
is about the same, i.e. an hour and a half; only 
that in the latter case the high water precedes the 
moon by that interval instead of following it. We 
may employ the same simple process at other 
places. Choose two days about a week distant ; 
find on each occasion the interval between the 
transit of the moon and the time of high water, 
then the mean of these two differences will always 
give some notion of the interval between high 
water and the moon's transit. If then we take 
from the Nautical Almanac the time of the moon's 
transit, and apply to it the correction proper for the 


port, we shall always have a sufficiently good tide- 
table to guide us in choosing a suitable time for 
taking our swim or our walk by the sea-side; 
though if you be the captain of a vessel, you will 
not be so imprudent as to enter port without 
taking counsel of the accurate tide-tables, for which 
we are indebted to the Admiralty. 

Every one who visits the sea-side, or who lives 
at a sea-port, should know this constant for the 
tides, which affect him and his movements so 
materially. If he will discover it from his own 
experience, so much the better. 

The first point to be ascertained is the time of 
high water. Do not take this from any local table ; 
you ought to observe it for yourself. You will go 
to the pier head, or, better still, to some place where 
the rise and fall of the mere waves of the sea will 
not embarrass you in your work. You must note 
by your watch the time when the tide is highest. 
An accurate way of doing this will be to have a 
scale on which you can measure the height at 
intervals of five minutes about the time of high 
water. You will then be able to conclude the 
time at which the tide was actually at its highest 


point ; but even if no great accuracy be obtainable, 
you can still get much interesting information, for 
you will without much difficulty be right within 
ten minutes or a quarter of an hour. 

The correction for the port is properly called the 
"establishment," this being the average time of 
high water on the days of full and change of the 
moon at the particular port in question. 

We can considerably amend the elementary 
notion of the tides which the former method has 
given us, if we adopt the plan described by Dr. 
Whewell in the first four editions of the Admiralty 
Manual of Scientific Inquiry. We speak of the 
interval between the transit of the moon and the 
time of high water as the luni-tidal interval. Of 
course at full and change this is the same thing as 
the establishment, but for other phases of the moon 
the establishment must receive a correction before 
being used as the luni-tidal interval. The correction 
is given by the following table 

Hour of Moon's transit after Sun : 







8 | 9 

20 m 

30 m 

50 m 

60 m 


60 m 

4om lom' + iom 


+10 m 

Correction of establishment to find luni-tidal interval : 


Thus at a port where the establishment was 3h. 
2501., let us suppose that the transit of the moon 
took place at 6 P.M. ; then we correct the establish- 
ment by 6om., and find the luni-tidal interval to 
be 2h. 25m., and accordingly the high water takes 
place at 8h. 2501. P.M. 

But even this method is only an approximation. 
The study of the tides is based on accurate observ- 
ation of their rise and fall on different places 
round the earth. To show how these observations 
are to be made, and how they are to be discussed 
and reduced when they have been made, I may 
refer to the last edition of the Admiralty Manual 
of Scientific Inquiry, 1886. For a complete study of 
the tides at any port a self-registering tide-gauge 
should be erected, on which not alone the heights 
and times of high and low water should be depicted, 
but also the continuous curve which shows at any 
time the height of the water. In fact, the whole 
subject of the practical observation and discussion 
and prediction of tides is full of valuable instruction, 
and may be cited as one of the most complete 
examples of the modern scientific methods. 

In the first place, the tide-gauge itself is a 


delicate instrument ; it is actuated by a float which 
rises and falls with the water, due provision being 
made that the mere influence of waves shall not 
make it to oscillate inconveniently. The motion of 
the float when suitably reduced by mechanism 
serves to guide a pencil, which, acting on the paper 
round a revolving drum, gives a faithful and un- 
intermitting record of the height of the water. 

Thus what the tide-gauge does is to present to 
us a long curved line of which the summits corre- 
spond to the heights of high water, while the 
depressions are the corresponding points of low 
water. The long undulations of this curve are, how- 
ever, very irregular. At spring tides, when the sun 
and the moon conspire, the elevations rise much 
higher and the depressions sink much lower than 
they do at neap tides, when the high water raised 
by the moon is reduced by the action of the sun. 
There are also many minor irregularities which 
show the tides to be not nearly such simple phe- 
nomena as might be at first supposed. But what 
we might hastily think of as irregularities are, in 
truth, the most interesting parts of the whole 
phenomena. Just as in the observations of the 


planets the study of- the perturbations has led 
us to results of the widest interest and instruction, 
so it is these minor phenomena of the tides which 
seem most pregnant with scientific interest. 

The tide-gauge gives us an elaborate curve. How 
are we to interpret that curve ? Here a beautiful 
mathematical theorem fortunately comes to our 
aid. Just as ordinary sounds consist of a number 
of undulations blended together, so the tidal wave 
consists of a number of distinct undulations 
superposed. Of these the ordinary lunar tide and 
the ordinary solar tide are the two principal ; but 
there are also minor undulations, harmonics, so 
to speak, some originating from the moon, some 
originating from the sun, and some from both 
bodies acting in concert. 

In the study of sound we can employ an 
acoustic apparatus for the purpose of decom- 
posing any proposed note, and finding not only the 
main undulation itself, but the several superposed 
harmonics which give to the note its timbre. 
So also we can analyze the undulation of the 
tide, and show the component parts. The de- 
composition is effected by the process known as 


harmonic analysis. The principle of the method 
may be very simply described. Let us fix our atten- 
tion on any particular "tide," for so the various 
elements are denoted. We can always determine 
beforehand, with as much accuracy as we may 
require, what the period of that tide will be. For 
instance, the period of the lunar semi-diurnal tide 
will of course be half the average time occupied 
by the moon in travelling round from the meridian 
of any place until it regains the same meridian ; 
the period of the lunar diurnal tide will be 
double as great; and there are fortnightly tides, 
and others of periods still greater. The essential 
point to notice is, that the periods of these tides 
are given by purely astronomical considerations, 
and do not depend upon the actual observations of 
the height of the water. 

We measure off on the curve the height of the 
tide at intervals of an hour. The larger the 
number of such measures that are available the 
better ; but even if there be only three hundred 
and sixty or seven hundred and twenty consecu- 
tive hours, then, as shown by Professor G. H. 
Darwin in the Admiralty Manual already referred 


to, it will still be possible to obtain a very com- 
petent knowledge of the tides in the particular 
port where the gauge has been placed. 

The art (for such indeed it may be described) of 
harmonic analysis consists in deducing from the 
hourly observations the facts with regard to each 
of the constituent tides. This art has been carried 
to such perfection, that it has been reduced to a 
very simple series of arithmetical operations. 
Indeed it has now been found possible to call in 
the aid of ingenious mechanism, by which the 
labours of computation are entirely superseded. 
The pointer of the harmonic analyzer has merely 
to be traced over the curve which the tide-gauge 
has drawn, and it is the function of the machine 
to decompose the composite undulation into its 
parts, and to exhibit the several constituent tides 
whose confluence gives the total result. 

As if nothing should be left to complete the 
perfection of a process which, both from its theo- 
retical and its practical sides, is of such importance, 
a machine for predicting tides has been designed, 
constructed, and is now in ordinary use. When 

by the aid of the harmonic analysis the effectiveness 

c 2 


of the several constituent tides affecting a port 
have become fully determined, it is of course 
possible to predict the tides for that port. Each 
" tide" is a simple periodic rise and fall, and we can 
compute for any future time the height of each 
were it acting alone. These heights can all be 
added together, and thus the height of the water 
is obtained. In this way a tide-table is formed, 
and such a table when complete will express not 
alone the hours and heights of high water on every 
day, but the height of the water at any intervening 

The computations necessary for this purpose are 
no doubt simple, so far as their principle is con- 
cerned ; but they are exceedingly tedious, and 
any process must be welcomed which affords 
mitigation of a task so laborious. The theory of 
the tides owes much to Sir William Thomson in 
the methods of observation and in the methods of 
reduction. He has now completed the practical 
parts of the subject by inventing and constructing 
the famous tide-predicting engine. 

The principle of this engine is comparatively 
simple. There is a chain which at one end is 

tlME AND TIDE. 3? 

fixed, and at the other end carries the pencil which 
is pressed against the revolving drum on which 
the prediction is to be inscribed. Between its 
two ends the chain passes up and down over 
pulleys. Each pulley corresponds to one of the 
" tides," and there are about a dozen altogether, 
some of which exercise but little effect. Of course 
if the centres of the pulleys were all fixed the pen 
could not move, but the centre of each pulley 
describes a circle with a radius proportional to the 
amplitude of the corresponding tide, and in a 
time proportional to the period of that tide. When 
these pulleys are all set so as to start at the proper 
phases, the motion is produced by turning round 
a handle which sets all the pulleys in motion, and 
makes the drum rotate. The tide curve is thus 
rapidly drawn out ; and so expeditious is the 
machine, that the tides of a port for an entire year 
can be completely worked out in a couple of 

While the student or the philosopher who seeks 
to render any account of the tide on dynamical 
grounds is greatly embarrassed by the difficulties 
introduced by friction, we, for our present purpose 


in the study of the great romance of modern 
science opened up to us by the theory of the tides, 
have to welcome friction as the agent which gives 
to the tides their significance from our point of 

There is a considerable difference between the 
height of the rise and fall of the tide at different 
localities. Out in mid-ocean, for instance, an island 
like St. Helena is washed by a tide only about 
three feet in range ; an enclosed sea like the 
Caspian is subject to no appreciable tides whatever, 
while the Mediterranean, notwithstanding its con- 
nection with the Atlantic, is still only subject to 
very inconsiderable tides, varying from one foot to 
a few feet. The statement that water always finds 
its own level must be received, like many another 
proposition in nature, with a considerable degree of 
qualification. Long ere one tide could have found 
its way through the Straits of Gibraltar in sufficient 
volume to have appreciably affected the level of the 
great inland sea, its effects would have been obli- 
terated by succeeding tides. On the other hand, 
there are certain localities which expose a funnel- 
shape opening to the sea ; into these the great tidal 


wave rushes, and as it passes onwards towards the 
narrow part, the waters become piled up so as to 
produce tidal phenomena of abnormal proportions. 
Thus, in our own islands, we have in the Bristol 
Channel a wide mouth into which a great tide 
enters, and as it hurries up the Severn it produces 
the extraordinary phenomenon of the Bore. The 
Bristol Channel also concentrates the great wave 
which gives Chepstow and Cardiff a tidal range of 
thirty-seven or thirty-eight feet at springs, and 
forces the sea up the river Avon so as to give 
Bristol a wonderful tide. There is hardly any 
more interesting spot in our islands for the observ- 
ation of tides than is found on Clifton Suspension 
Bridge. From that beautiful structure you look 
down on a poor and not very attractive stream, 
which two hours later becomes transformed into a 
river of ample volume, down which great ships are 
navigated. But of all places in the world, the most 
colossal tidal phenomena are those in the Bay of 
Fundy. Here the Atlantic passes into a long 
channel whose sides gradually converge. When 
the great pulse of the tide rushes up this channel, 
it is gradually accumulated into a mighty volume 


at the upper end, the ebb and flow of which at 
spring tides extends through an astonishing range 
of more than fifty feet, 

The discrepancies between the tides at different 
places are chiefly due to the local formations 
of the coasts and the sea-beds. Indeed, it seems 
that if the whole earth were covered with an 
uniform and deep ocean of water, the tides would 
be excessively feeble. On no other supposition 
can we reasonably account for the fact that our 
barometric records fail to afford us any very 
distinct evidence as to the existence of tides in 
the atmosphere. For you will, of course, remem- 
ber that our atmosphere may be regarded as a deep 
and vast ocean of air, which embraces the whole 
earth, extending far above the loftiest summits of 
the mountains. 

It is one of the profoundest of nature's laws that 
wherever friction takes place, energy has to be 
consumed. Perhaps I ought rather to say trans- 
formed, for of course it is now well known that 
consumption of energy in the sense of absolute 
loss is impossible. Thus, when energy is expended 
in moving a body in opposition to the force of 


friction, or in agitating a liquid, the energy which 
disappears in its mechanical form reappears in the 
form of heat. The agitation of water by paddles 
moving through it warms the water, and the 
accession of heat thus acquired measures the 
energy which has been expended in making the 
paddles rotate. The motion of a liquid of which 
the particles move among each other with friction, 
can only be sustained by the incessant degradation 
of energy from the mechanical form into the lower 
form of diffused heat. Thus the very fact that the 
tides are ebbing and flowing, and that there is 
consequently incessant friction going on among 
all the particles of water in the ocean, shows us 
that there must be some great store of energy 
constantly available to supply the incessant 
draughts made upon it by the daily oscillation 
of the tides. In addition to the mere friction 
between the particles of water, there are also many 
other ways in which the tides proclaim to us that 
there is some great hoard of energy which is 
continually accessible to their wants. Stand on 
the bank of an estuary or river up and down 
which a great tidal current ebbs and flows ; you 


will see the water copiously charged with sediment 
which the tide is bearing along. Engineers are 
well aware of the potency of the tide as a vehicle 
for transporting stupendous quantities of sand or 
mud. A sand-bank impedes the navigation of a 
river; the removal of that sand-bank would be a 
task, perhaps, conceivably possible by the use of 
steam dredges and other appliances, whereby vast 
quantities of sand could be raised and transported 
to where they can be safely deposited in deep water. 
It is sometimes possible to effect the desired end 
by applying the power of the tide. A sea-wall 
judiciously thrown out can be made to concentrate 
the tide into a much narrower channel. Its daily 
oscillations will be accomplished with greater 
vehemence, and as the tide rushes furiously back- 
wards and forwards over the obstacle, the incessant 
action will gradually remove it, and the impedi- 
ment to navigation may be cleared away. Here we 
actually see the tides performing a piece of definite 
and very laborious work, to accomplish which by the 
more ordinary agents would be a stupendous task. 
In some places the tides are actually harnessed 
so as to accomplish useful work. I have read that 


underneath old London Bridge there used formerly 
to be great water-wheels, which were turned by 
the tide as it rushed up the river, and turned again, 
though in the opposite way, by the ebbing tide. 
These wheels were, I believe, employed to pump up 
water, though it does not seem obvious for what 
purposes the water would have been suitable. 
Indeed in the ebb and flow all round our coasts 
there is a potential source of energy which has 
generally been allowed to run to waste, save per- 
haps in one or two places in the south of England. 
The tide could be utilized in various ways. Many 
of us have noticed the floating mills on the Rhine. 
They are vessels like paddle steamers anchored in 
the rapid current. The flow of the river makes the 
paddles rotate, and thus the machinery in the 
interior is worked. Such craft moored in a rapid 
tide-way could also be made to convey the power 
of the tides into the mechanism of the mill. Or there 
is still another method which has been employed, 
and which will perhaps have a future before it in 
those approaching times when the coal-cellars of 
England shall be exhausted. Imagine on the sea- 
coast a large flat extent which is inundated twice 
every day by the tide. Let us build a stout 


wall round this area, and provide it with a sluice- 
gate. Open the gate as the tide rises, and the 
great pond will be filled ; then at the moment of 
high water close the sluice, and the pond-full will 
be impounded. If at low tide' the sluice be opened 
the water will rush tumultuously out. Now suppose 
that a water-wheel be provided, so that the rapid 
rush of water from the exit shall fall upon its 
blades ; then a source of power is obviously the 

At present, however, such a contrivance would 
find but few advocates, for of course the com- 
mercial aspect of the question is that which will 
decide whether the scheme is practicable and 
economical. The issue indeed can be very simply 
stated. Suppose that a given quantity of power be 
required let us say that of one hundred horse. Then 
we have to consider the conditions under which a 
contrivance of the kind we have sketched shall yield 
a power of this amount. Sir William Thomson, 
in a very interesting address to the British Asso- 
ciation at York in 1881, discussed this question, 
and I shall here make use of the facts he brought 
forward on that occasion. He showed that to 


obtain as much power as could be produced by a 
steam-engine of one hundred horse power, a very 
large reservoir would be required. It is doubtful 
indeed whether there would be many localities on 
the earth which would be suitable for the purpose. 
Suppose, however, an estuary could be found 
which had an area of forty acres ; then if a wall 
were thrown across the mouth so that the tide 
could be impounded, the total amount of power 
that could be yielded by a water-wheel worked by 
the incessant influx and efflux of the tide would be 
equal to that yielded by the one hundred horse 
engine, running continuously from one end of the 
year to the other. 

There are many drawbacks to a tide-mill of 
this description. In the first place, its situation 
would naturally be far removed from other con- 
veniences necessary for manufacturing purposes. 
Then too there is the great irregularity in the 
way in which the power is rendered available. At 
certain periods during the twenty-four hours the 
mill would stop running, and the hours when this 
happened would be constantly changing. The 
inconvenience from the manufacturer's point of 


view of a deficiency of power during neap tides 
might not be compensated by the fact that he had 
an excessive supply of power at spring-tides. 
Before tide-mills could be suitable for manufactur- 
ing purposes, some means must be found for storing 
away the energy when it is redundant, and apply- 
ing it when its presence is required. We should 
want in fact for great sources of energy some con- 
trivance which should fulfil the same function as 
the accumulators do in an electrical installation. 

Even then, however, the financial consideration 
remains, as to whether the cost of building the dam 
and maintaining the tide-mill in good order will 
not on the whole exceed the original price and 
the charges for the maintenance of a hundred 
horse power steam-engine. There cannot be a 
doubt that in this epoch of the earth's history, 
so long as the price of coal is only a few shillings a 
ton, the tide-mill, even though we seem to get its 
power without current expense, is vastly more ex- 
pensive than a steam-engine. Indeed, Sir William 
Thomson remarks, that wherever a suitable tidal 
basin could be found, it would be nearly as easy to 
reclaim the land altogether from the sea. And if 


this were in any locality where manufactures were 
possible, the commercial value of forty acres of re- 
claimed land would greatly exceed all the expenses 
attending the steam-engine. But when the time 
comes, as come it apparently will, that the price of 
coal shall have risen to several pounds a ton, the 
economical aspect of steam as compared with other 
prime movers will be greatly altered ; it will then 
no doubt be found advantageous to utilize great 
sources of energy, such as Niagara and the tides, 
which it is now more prudent to let run to waste. 

For my argument, however, it matters little that 
the tides are not constrained to do much useful 
work. They are always doing work of some kind, 
whether that be merely heating the particles of 
water by friction, or vaguely transporting sand 
from one part of the ocean to the other. Useful 
work or useless work are alike from the present 
point of view. We know that work can never be 
done unless by the consumption or transformation 
of energy. For each unit of work that is done 
whether by any machine or contrivance, by the 
muscles of man or any other animal, by the winds, 
the waves, or the tides, or in any other way 


whatever a certain equivalent quantity of energy 
must have been expended. When, therefore, we see 
any work being performed, we may always look for 
the source of energy to which the machine owes its 
efficiency. Every machine illustrates the old story, 
that perpetual motion is impossible. A mechanical 
device, however ingenious may be the construction, 
or however accurate the workmanship, can never 
possess what is called perpetual motion. It is 
needless to enter into details of any proposed con- 
trivance of wheels, of pumps, of pulleys ; it is 
sufficient to say that nothing in the shape of me- 
chanism can work without friction, that friction 
produces heat, that heat is a form of energy, and 
that to replace the energy consumed in producing 
the heat there must be some source from which 
the machine is replenished if its motion is to be 
continued indefinitely. 

Hence, as the tides may be regarded as a machine 
doing work, we have to ascertain the origin of that 
energy which they are continually expending. It 
is at this point that we first begin to feel the diffi- 
culties inherent in the theory of tidal evolution. I 
do not mean difficulties in the sense of doubts, for 


up to the present I have mentioned no doubtful 
point. When I come to such I shall give due 
warning. By difficulties I now mean points which 
it is not easy to understand without a little dynam- 
ical theory ; but we must face these difficulties, 
and endeavour to elucidate them as well as we 

Let us first see what the sources of energy can 
possibly be on which the tides are permitted to 
draw. Our course is simplified by the fact that 
the energy of which we have to speak is of a 
mechanical description, that is to say, not in- 
volving heat or other more obscure forms of 
energy. A simple type of energy is that pos- 
sessed by a clock-weight after the clock has been 
wound. A store of power is thus laid up which 
is gradually doled out during the week in small 
quantities, second by second, to sustain the 
motion of the pendulum. The energy in this case 
is due to the fact that the weight is attracted by 
the earth, and is yielded according as the weight 
sinks downwards. In the separation between two 
mutually attracting bodies, a store of energy is 
thus implied. What we learn from an ordinary 



clock may be extended to the great bodies of the 
universe. The moon is a gigantic globe separated 
from our earth by a distance of 240,000 miles. 
The attraction between these two bodies always 
tends to bring them together. No doubt the 
moon is not falling towards the earth as the de- 
scending clock- weight is doing. We may, in fact, 
consider the moon, so far as our present object is 
concerned, to be revolving almost in a circle, of 
which the earth is the centre. If the moon, how- 
ever, were to be stopped, it would at once com- 
mence to rush down towards the earth, whither it 
would arrive with an awful crash in the course of 
four or five days. It is fortunately true that the 
moon does not behave thus ; but it has the ability 
of doing so, and thus the mere separation between 
the earth and the moon involves the existence of 
a stupendous quantity of energy, capable under 
certain conditions of undergoing transformation. 

There is also another source of mechanical energy 
besides that we have just referred to. A rapidly 
moving body possesses, in virtue of its motion, a 
store of readily available energy, and it is easy to 
show that energy of this type is capable of trans- 


formation into other types. Think of a cannon- 
ball rushing through the air at a speed of a thousand 
feet per second ; it is capable of wreaking disaster 
wherever its blow may fall, simply because its 
rapid motion is the vehicle by which the energy 
of the gunpowder is transferred from the gun to 
where the blow is to be struck. Had the cannon 
been directed vertically upwards, then the pro- 
jectile, leaving the muzzle with the same initial 
velocity as before, would soar up and up, with 
gradually abating speed, until at last it reached a 
turning-point, the elevation of which would depend 
upon the initial velocity. Poised for a moment at 
the summit, the cannon-ball may then be likened 
to the clock-weight, for the entire energy which 
it possessed by its motion has been transformed 
into the statical energy of a raised weight. Thus 
we see these two forms of energy are mutually 
interchangeable. The raised weight if allowed to 
fall will acquire velocity, or the rapidly moving 
weight if directed upwards will acquire altitude. 
The quantity of energy which can be conveyed 
by a rapidly moving body increases greatly with 
its speed. For instance, if the speed of the body 

D 2 


be doubled, the energy will be increased fourfold, 
or, in general, the energy which a moving body 
possesses may be said to be proportional to the 
square of its speed. Here then we have another 
source of the energy present in our earth-moon 
system ; for the moon is hurrying along in its path 
with a speed of two-thirds of a mile per second, 
or about twice or three times the speed of a cannon- 
shot. Hence the fact that the moon is continu- 
ously revolving round the earth shows us that 
it possesses a store of energy which is nine times 
as great as that which a cannon-ball as massive 
as the moon, and fired with the ordinary velocity, 
would receive from the powder which discharged it. 
Thus we see that the moon is endowed with 
two sources of energy, one of which is due to its 
separation from the earth, and the other to the 
speed of its motion. Though these are distinct, 
they are connected together by a link which it is 
important for us to comprehend. The speed with 
which the moon revolves around the earth is con- 
nected with the moon's distance from the earth. 
The moon might, for instance, revolve in a larger 
circle than that which it actually pursues ; but if 


it did so, the speed of its motion would have to be 
appropriately lessened. The orbit of the moon 
might have a much smaller radius than it has at 
present, provided that the speed was sufficiently 
increased to compensate for the increased at- 
traction which the earth would exercise at the 
lessened distance. Indeed, I am here only stating 
what every one is familiar with under the form of 
Kepler's Law, that the square of the periodic time 
is in proportion to the cube of the mean distance. 
To each distance of the moon therefore belongs an 
appropriate speed. The energy due to the moon's 
position and the energy due to its motion are 
therefore connected together. One of these quan- 
tities cannot be altered without the other under- 
going change. If the moon's orbit were increased 
there would be a gain of energy due to the 
enlarged distance, and a loss of energy due to the 
diminished speed. These would not, however, 
exactly compensate. On the whole, we may 
represent the total energy of the moon as a single 
quantity, which increases when the distance of 
the moon from the earth increases, and lessens 
when the distance from the earth to the moon 


lessens. For simplicity we may speak of this as 

But the most important constituent of the store 
of energy in the earth-moon system is that con- 
tributed by the earth itself. I do not now speak 
of the energy due to the velocity of the earth in 
its orbit round the sun. The moon indeed partici- 
pates in this equally with the earth, but it does 
not affect those mutual actions between the earth 
and moon with which we are at present concerned. 
We are, in fact, discussing the action of that piece 
of machinery the earth-moon system ; and its 
action is not affected by the circumstance that 
the entire machine is being bodily transported 
around the sun in a great annual voyage. This 
has little more to do with our present argument 
than has the fact that a man is walking about to 
do with the motions of the works of the watch in 
his pocket. We shall, however, have to allude to 
this subject further on. 

The energy of the earth which is significant in 
the earth-moon theory is due to the earth's rota- 
tion upon its axis. We may here again use as 
an illustration the action of machinery ; and the 


special contrivance that I now refer to is the 
punching-engine that is used in our ship-building 
works. In preparing a plate of iron to be riveted 
to the side of a ship, a number of holes have to be 
made all round the margin of the plate. These 
holes must be half an inch or more in diameter, 
and the plate is sometimes as much as, or more 
than, half an inch in thickness. The holes are pro- 
duced in the metal by forcing a steel punch through 
it ; and this is accomplished without even heating 
the plate so as to soften the iron. It is needless to 
say that an intense force must be applied to the 
punch. On the other hand, the distance through 
which the punch has to be moved is comparatively 
small. The punch is attached to the end of a 
powerful lever, the other end of the lever is raised 
by a cam, so as to depress the punch to do its 
work. An essential part of the machine is a small 
but heavy fly-wheel connected by gearing with 
the cam. 

This fly-wheel when rapidly revolving con- 
tains within it, in virtue of its motion, a large 
store of energy which has gradually accumulated 
during the time that the punch is not in action. 


The energy is no doubt originally supplied from 
a steam-engine. What we are especially concerned 
with is the action of the rapidly rotating wheel as 
a reservoir in which a large store of energy can be 
conveniently maintained until such time as it is 
wanted. In the action of punching, when the 
steel die comes down upon the surface of the plate, 
a large quantity of energy is suddenly demanded 
to force the punch against the intense resistance 
it experiences ; the energy for this purpose is drawn 
from the store in the fly-wheel, which experi- 
ences no doubt a check in its velocity, to be re- 
stored again from the energy of the engine during 
the interval which elapses before the punch is 
called on to make the next hole. 

Another illustration of the fly-wheel on a splen- 
did scale is to be seen in steel works, where 
railway lines are being manufactured. A white- 
hot ingot of steel is presented to a pair of power- 
ful rollers, which grip the steel, and send it through 
at the other side both compressed and elongated. 
Tremendous power is required to meet the sudden 
demand on the machine at the critical moment. 
To obtain this power an engine of immense 


horse-power is sometimes attached directly to the 
rollers, but more frequently an engine of rather less 
horse-power will be used, the might of this engine 
being applied to giving rapid rotation to a fly- 
wheel, which may thus be regarded as a reservoir 
full of energy. The rolling mills then obtain from 
this store in the fly-wheel whatever energy is 
necessary for their gigantic task 

These illustrations will suffice to show how a 
rapidly rotating body may contain energy in 
virtue of its rotation, just as a cannon-ball contains 
energy in virtue of its speed of translation, or as 
a clock-weight has energy in virtue of the fact 
that it has some distance to fall before it reaches 
the earth. The rotating body need not necessarily 
have the shape of a wheel it may be globular in 
form ; nor need the axes of rotation be fixed in 
bearings, like those of the fly-wheel ; nor of course 
is there any limit to the dimensions which the 
rotating body may assume. Our earth is, in fact, 
a vast rotating body 8000 miles in diameter, and 
turning round upon its axis once every twenty- 
three hours and fifty-six minutes. Viewed in this 
way, the earth is to be regarded as a gigantic 


fly-wheel containing a quantity of energy great in 
correspondence with the earth's mass. The amount 
of energy which can be stored by rotation also 
depends upon the square of the velocity with 
which the body turns round ; thus if our earth 
turned round in half the time which it does at 
present, that is, if the day was twelve hours instead 
of twenty-four hours, the energy contained in 
virtue of that rotation would be four times its 
present amount. 

Reverting now to the earth-moon system, the 
energy which that system contains consists essen- 
tially of two parts the moon-energy, whose compo- 
site character I have already explained, and the 
earth-energy, which has its origin solely in the 
rotation of the earth on its axis. It is necessary 
to observe that these are essentially distinct there 
is no necessary relation between the speed of 
the earth's rotation and the distance of the moon, 
such as there is between the distance of the moon 
and the speed with which it revolves in its orbit. 

For completeness, it ought to be added that 
there is also some energy due to the moon's rota- 
tion on its axis, but this is very small for two 


reasons : first, because the moon is small compared 
with the earth, and second, because the angular 
velocity of the moon is also very small compared 
with that of the earth. We may therefore dismiss 
as insignificant the contributions from this source 
to the sum total of energy. 

I have frequently used illustrations derived from 
machinery, but I want now to emphasize the 
profound distinction that exists between the rotation 
of the earth and the rotation of a fly-wheel in 
a machine shop. They are both, no doubt, energy- 
holders, but it must be borne in mind, that as the 
fly-wheel doles out its energy to supply the wants 
of the machines with which it is connected, a resti- 
tution of its store is continually going on by the 
action of the engine, so that on the whole the 
speed of the fly-wheel does not slacken. The 
earth, however, must be likened to a fly-wheel 
which has been disconnected with the engine. If, 
therefore, the earth have to supply certain demands 
on its accumulation of energy, it can only do so 
by a diminution of its hoard, and this involves a 
sacrifice of some of its speed. 

In the earth-moon system there is no engine 


at hand to restore the losses of energy which are 
inevitable when work has to be done. But we have 
seen that work is done ; we have shown, in fact, 
that the tides are at present doing work, and have 
been doing work for as long a period in the 
past as our imagination can extend to. The 
energy which this work has necessitated can only 
have been drawn from the existing store in the 
system ; that energy consists of two parts the 
moon-energy and the earth's rotation energy. The 
problem therefore for us to consider is, which of 
these two banks the tides have drawn on to meet 
their constant expenditure. This is not a question 
that can be decided offhand ; in fact, if we attempt 
to decide it in an offhand manner we shall 
certainly go wrong. It seems so very plausible 
to say that as the moon causes the tides, therefore 
the energy which these tides expend should be 
contributed by the moon. But this is not the case. 
It actually happens that though the moon does 
cause the tides, yet when those tides consume 
energy they draw it not from the distant moon, 
but from the vast supply which they find ready 
to their hand, stored up in the rotation of the earth. 


The demonstration of this is not a very simple 
matter ; in fact, it is so far from being simple that 
many philosophers, including some eminent ones 
too, while admitting that of course the tides must 
have drawn their energy from one or other or both 
of these two sources, yet found themselves unable 
to assign how the demand was distributed between 
the two conceivable sources of supply. 

We are indebted to Professor Purser of Belfast 
for having indicated the true dynamical principle 
on which the problem depends. It involves 
reasoning based simply on the laws of motion and 
on elementary mathematics, but not in the least 
involving questions of astronomical observation. 
It would be impossible for me in a lecture like 
this to give any explanation of the mathematical 
principles referred to. I shall, however, endeavour 
by some illustrations to set before you what this 
profound principle really is. Were I to give it the 
old name I should call it the law of the conserv- 
ation of areas ; the more modern writers, however, 
speak of it as the conservation of moment of mo- 
mentum, an expression which exhibits the nature 
of the principle in a more definite manner. 


I do not see how to give any very accurate 
illustration of what this law means, but I must 
make the attempt, and if you think the illustration 
beneath the dignity of the subject, I can only 
plead the difficulty of mathematics as an excuse. 
Let us suppose that a ball-room is fairly filled 
with dancers, or those willing to dance, and that 
a merry waltz is being played ; the couples have 
formed, and the floor is occupied with pairs who 
are whirling round and round in that delightful 
amusement. Some couples drop out for a while 
and others strike in ; the fewer couples there are 
the wider is the range around which they can 
waltz, the more numerous the couples the less 
individual range will they possess. I want you to 
realize that in the progress of the dance there is 
a certain total quantity of spin at any moment 
in progress ; this spin is partly made up of the 
rotation by which each dancer revolves round his 
partner, and partly of the circular orbit about the 
room which each couple endeavours to describe. 
If there are too many couples on the floor for the 
general enjoyment of the dance, then both the orbit 
and the angular velocity of each couple will be 


restricted by the interference with their neighbours. 
We may, however, assert that so long as the dance 
is in full swing the total quantity of spin, partly 
rotational and partly orbital, will remain constant. 
When there are but few couples the unimpeded 
rotation and the large orbits will produce as much 
spin as when there is a much larger number of 
couples, for in the latter case the diminished 
freedom will lessen the quantity of spin produced 
by each individual pair. It will sometimes happen 
too that collision will take place, but the slight 
diversions thus arising only increase the general 
merriment, so that the total quantity of spin may 
be sustained, even though one or two couples 
are placed temporarily liors de combat. I have 
invoked a ball-room for the purpose of bringing 
out what we may call the law of the conservation 
of spin. No matter how much the individual per- 
formers may change, or no matter what vicissi- 
tudes arise from their collision and other mutual 
actions, yet the total quantity of spin remains 

Let us look at the earth-moon system. The 
law of the conservation of moment of momentum 


may, with sufficient accuracy for our present 
purpose, be interpreted to mean that the total 
quantity of spin in the system remains unaltered. 
In our system the spin is threefold ; there is 
first the rotation of the earth on its axis, there is 
the rotation of the moon on its axis, and then 
there is the orbital revolution of the moon around 
the earth. The law to which we refer asserts that 
the total quantity of these three spins, each esti- 
mated in the proper way, will remain unaltered. 
It matters not that tides may ebb and flow, or that 
the distribution of the spin shall vary, but its total 
amount remains inflexibly constant. One con- 
stituent of the total amount that is, the rotation 
of the moon on its axis is so insignificant, that 
for our present purposes it may be entirely disre- 
garded. We may therefore assert that the amount 
of spin in the earth, due to its rotation round its 
axis, added to the amount of spin in the moon 
due to its revolution round the earth, remains 
unalterable. If one of these quantities change by 
increase or by decrease, the other must correspond- 
ingly change by decrease or by increase. If, there- 
fore, from any cause, the earth began to spin a little 


more quickly round its axis, the moon must do a 
little less spin ; and consequently, it must shorten 
its distance from the earth. Or suppose that the 
earth's velocity of rotation is abated, then its con- 
tribution to the total amount of spin is lessened ; 
the deficiency must therefore be made up by the 
moon, but this can only be done by an enlarge- 
ment of the moon's orbit. I should add, as a 
caution, that these results are true only on the 
supposition that the earth-moon system is isolated 
from all external interference. With this proviso, 
however, it matters not what may happen to the 
earth or moon, or what influence one of them may 
exert upon the other. No matter what tides may 
be raised, no matter even if the earth fly into 
fragments, the whole quantity of spin of all those 
fragments would, if added to the spin of the 
moon, yield the same unalterable total. We are 
here in possession of a most valuable dynamical 
principle. We are not concerned with any special 
theory as to the action of the tides ; it is sufficient 
for us that in some way or other the tides have 
been caused by the moon, and that being so, the 
principle of the conservation of spin will apply. 



Were the earth and the moon both rigid bodies, 
then there could be of course no tides on the 
earth, for if rigid it is devoid of ocean. The 
rotation of the earth on its axis v/ould therefore 
be absolutely without change, and therefore the 
necessary condition of the conservation of spin 
would be very simply attained by the fact that 
neither of the constituent parts changed. The 
earth, however, not being entirely rigid, and being 
subject to tides, this simple state of things cannot 
continue ; there must be some change in progress. 

I have already shown that the fact of the 
ebbing and the flowing of the tide necessitates an 
expenditure of energy, and we saw that this energy 
must come either from that stored up in the earth 
by its rotation, or from that possessed by the 
moon in virtue of its distance and revolution. The 
law of the conservation of spin will enable us to 
decide at once as to whence the tides get their 
energy. Suppose they took it from the moon, the 
moon would then lose in energy, and consequently 
come nearer the earth. The quantity of spin con- 
tributed by the moon would therefore be lessened, 
and accordingly the spin to be made up by the 


earth would be increased. That means, of course, 
that the velocity of the earth rotating on its axis 
must be increased, and this again would necessi- 
tate an increase in the earth's rotational energy. 
It can be shown, too, that to keep the total spin 
right, the energy of the earth would have to gain 
more than the moon would have lost by revolving 
in a smaller orbit. Thus we find that the total 
quantity of energy in the system would be increased. 
This would lead to the absurd result that the action 
of the tides manufactured energy in our system. 
Of course, such a doctrine cannot be true ; it would 
amount to a perpetual motion ! We might as 
well try to get a steam-engine which would 
produce enough heat by friction not only to 
supply its own boilers, but to satisfy the thermal 
wants of the whole parish. We must therefore 
adopt the other alternative. The tides do not 
draw their energy from the moon ; they draw it 
from the store possessed by the earth in virtue 
of its rotation. 

We can now state the end of this rather long 
discussion in a very simple and brief manner. 
Energy can only be yielded by the earth at the 

E 2 


expense of some of the speed of its rotation. The 
tides must therefore cause the earth to revolve 
more slowly ; in other words, the tides are increasing 
the length of the day. 

The earth therefore loses some of its velocity 
of rotation ; consequently it does less than its due 
share of the total quantity of spin, and an increased 
quantity of spin must therefore be accomplished 
by the moon ; but this can only be done by an 
enlargement of its orbit. Thus there are two great 
consequences of the tides in the earth-moon system 
the days are getting longer, the moon is receding 

These points are so important that I shall try 
and illustrate them in another way, which will 
show, at all events, that one and both of these tidal 
phenomena commend themselves to our common 
sense. Have we not shown how the tides in their 
ebb and flow are incessantly producing friction, 
and have we not also likened the earth to a great 


wheel? When the driver wants to stop a railway 
train the brakes are put on, and the brake is merely 
a contrivance for applying friction to the circum- 
ference of a wheel for the purpose of checking its 


motion. Or when a great weight is being lowered 
by a crane, the motion is checked by a band which 
applies friction on the circumference of a wheel 
arranged for the special purpose. Need we then 
be surprised that the friction of the tides acts like 
a brake on the earth, and gradually tends to check 
its mighty rotation ? The progress of lengthening 
the day by the tides is thus readily intelligible. It 
is not quite so easy to see why the ebbing and the 
flowing of the tide on the earth should actually 
have the effect of making the moon to retreat ; this 
phenomenon is in deference to a profound law of 
nature, which tells us that action and reaction are 
equal and opposite to each other. If I might 
venture on a very homely illustration, I may say 
that the moon, like a troublesome fellow, is con- 
stantly annoying the earth by dragging its waters 
backward and forward by means of tides ; and the 
earth, to free itself from this irritating interference, 
tries to push off the aggressor and to make him 
move further away. 

Another way in which we can illustrate the 
retreat of the moon as the inevitable consequence 
of tidal friction is shown in the adjoining figure, in 



which the large body 
E represents the earth, 
and the small body M 
the moon. We may for 
simplicity regard the 
moon as a point, and 
as this attracts each 
particle of the earth, 
the total effect of the 
moon on the earth may 
be represented by a 
single force. By the 
law of equality of action 
and reaction, the force 
of the earth on the 
moon is to be repre- 
sented by an equal and 
opposite force. If there 
were no tides then the 
moon's force would of 
course pass through the 
earth's centre ; but as the effect of the moon is to 
slacken the earth's rotation, it follows that the total 
force does not exactly pass through the line of the 



earth's centre, but a little to one side, in order to 
pull the opposite way to that in which the earth 
is turning, and thus bring down its speed. We 
may therefore decompose the earth's total force 
on the moon into two parts, one of which tends 
directly towards the earth's centre, while the other 
acts tangentially to the moon's orbit. The central 
force is of course the main guiding power which 
keeps the moon in its path ; but the incessant 
tangential force constantly tends to send the moon 
out further and further, and thus the growth of its 
orbit can be accounted for. 

We therefore conclude finally, that the tides are 
making the day longer and sending the moon away 
further. It is the development of the consequences 
of these laws that specially demands our attention 
in these lectures. We must have the courage to 
look at the facts unflinchingly, and deduce from 
them all the wondrous consequences they involve. 
Their potency arises from a characteristic feature 
they are unintermitting. Most of the great 
astronomical changes with which we are ordinarily 
familiar are really periodic : they gradually in- 
crease in one direction for years, for centuries, or 


for untold ages ; but then a change comes, and the 
increase is changed into a decrease, so that after 
the lapse of becoming periods the original state 
of things is restored. Such periodic phenomena 
abound in astronomy. There is the annual fluctu- 
ation of the seasons ; there is the eighteen or nine- 
teen year period of the moon ; there is the great 
period of the precession of the equinoxes, amount- 
ing to twenty-six thousand years ; and then there 
is the stupendous Annus Magnus of hundreds of 
thousands of years, during which the earth's orbit 
itself breathes in and out in response to the attrac- 
tion of the planets. But these periodic phenomena, 
however important they may be to us mere 
creatures of a day, are insignificant in their effects 
on the grand evolution through which the celestial 
bodies are passing. The really potent agents in 
fashioning the universe are those which, however 
slow or feeble they may seem to be, are inces- 
sant in their action. The effect which a cause 
is competent to produce depends not alone 
upon the intensity of that cause, but also upon 
the time during which it has been in operation. 
From the phenomena of geology, as well as from 


those of astronomy, we know that this earth and 
the system to which it belongs has endured for 
ages, not to be counted by scores of thousands of 
years, or, as Prof. Tyndall has so well remarked, 
" Not for six thousand years, nor for sixty thousand 
years, nor six hundred thousand years, but for 
aeons of untold millions." Those slender agents 
which have devoted themselves unceasingly to 
the accomplishment of a single task may in this 
long lapse of time have accomplished results 
of stupendous magnitude. In famed stalactite 
caverns we are shown a colossal figure of crystal 
extending from floor to roof, and the formation 
of that column is accounted for when we see a 
tiny drop falling from the roof above to the floor 
beneath. A lifetime may not suffice for that falling 
drop to add an appreciable increase to the stalactite 
down which it trickles, or to the growing stalagmite 
on which it falls ; but when the operation has been 
in progress for immense ages, it is capable of the 
formation of the stately column. Here we have an 
illustration of an influence which, though apparently 
trivial, acquires colossal significance when adequate 
time is afforded. It is phenomena of this kind 


which the student of nature should most narrowly 
watch, for they are the real architects of the 

The tidal consequences which we have already 
demonstrated are emphatically of this non-pe- 
riodic class the day is always lengthening, the 
moon is always retreating. To-day is longer than 
yesterday ; to-morrow will be longer than to-day. 
It cannot be said that the change is a great one; 
it is indeed too small to be appreciable even by 
our most delicate observations. In one thousand 
years the alteration in the length of a day is only 
a small fraction of a second ; but what may be a 
very small matter in one thousand years can be- 
come a very large one in many millions of years. 
Thus it is that when we stretch our view through 
immense vistas of time past, or when we look for- 
ward through immeasurable ages of time to come, 
the alteration in the length of the day will assume 
the most startling proportions, and involve the 
most momentous consequences. 

Let us first look back. There was a time when 
the day, instead of being the twenty-four hours we 
now have, must have been only twenty-three hours, 


How many millions of years ago that was I do 
not pretend to say, nor is the point material for 
our argument ; suffice it to say, that assuming, as 
geology assures us we may assume, the existence 
of these aeons of millions of years, there was once 
a time when the day was not only one hour 
shorter, but was even several hours less than it is 
at present. Nor need we stop our retrospect at a 
day of even twenty, or fifteen, or ten hours 
long ; we shall at once project our glance back to 
an immeasurably remote epoch, at which the earth 
was spinning round in a time only one sixth or 
even less of the length of the present day. There 
is here a reason for our retrospect to halt, for at 
some eventful period, when the day was about 
three or four hours long, the earth must have been 
in a condition of a very critical kind. 

It is well known that fearful accidents occasion- 
ally happen where large grindstones are being 
driven at a high speed. The velocity of rotation 
becomes too great for the tenacity of the stone to 
withstand the stress ; a rupture takes place, the 
stone flies in pieces, and huge fragments are 
hurled around. For each particular grindstone 


there is a certain special velocity depending upon 
its actual materials and character, at which it would 
inevitably fly in pieces. I have once before likened 
our earth to a wheel ; now let me liken it to a 
grindstone. There is therefore a certain critical 
velocity of rotation for the earth at which it would 
be on the brink of rupture. We cannot exactly 
say, in our ignorance of the internal constitution 
of the earth, what length of day would be the 
shortest possible for our earth consistently with 
the preservation of its integrity ; we may, how- 
ever, assume that it will be about three or four 
hours, or perhaps a little less than three. The 
exact amount, however, is not really very material; 
it would be sufficient for our argument to assert 
that there is a certain minimum length of day for 
which the earth can hold together. In our retro- 
spect, therefore, through the abyss of time our 
view must be bounded by that state of the earth 
when it is revolving in this critical period. With 
what happened before that we shall not at present 
concern ourselves. Thus we look back to a time at 
the beginning of the present order of things, when 
the day was only some three or four hours long. 


Let us now look at the moon, and examine 
where it must have been during these past ages. 
As the moon is gradually getting further and fur- 
ther from us at present, so, looking back into past 
time, we find that the moon was nearer and nearer 
to the earth the further back our view extends. 
If we concentrate our attention solely on essen- 
tial features, we may say that the path of the 
moon is a sort of spiral which winds round and 
round the earth, gradually getting larger, though 
with extreme slowness. Looking back we see this 
spiral gradually coiling in and in, until in a retro- 
spect of millions of years, instead of its distance 
from the earth being 240,000 miles, it must have 
been much less. There was a time when the 
moon was only 200,000 miles away ; there was 
a time many millions of years ago, when the moon 
was only 100,000 miles away. Nor need we here 
stop our argument ; we may look further and 
further back, and follow the moon's spiral path as 
it creeps in and in towards the earth, until at last 
it appears actually in contact with that great globe 
of ours, from which it is now separated by a quarter 
of a million of miles. 


Surely the tides have thus led us to the know- 
ledge of an astounding epoch in our earth's past 
history, when the earth is spinning round in a few 
hours, and when the moon is, practically speaking, 
in contact with it. Perhaps I should rather say, 
that the materials of our present moon were in this 
situation, for we would hardly be entitled to assume 
that the moon then possessed the same globular 
form in which we see it now. To form a just 
apprehension of the true nature of both bodies at 
this critical epoch, we must study their concurrent 
history as it is disclosed to us by a totally different 
line of reasoning. 

Drop, then, for a moment all thought of tides, 
and let us bring to our aid the laws of heat, which 
will disclose certain facts in the ancient history 
of the earth-moon system perhaps as astounding 
as those to which the tides have conducted us. 
In one respect we may compare these laws of heat 
with the laws of the tides ; they are both alike 
non-periodic, their effects are cumulative from age 
to age, and imagination can hardly even impose 
a limit to the magnificence of the works they can 
accomplish. Our argument from heat is founded 


on a very simple matter. It is quite obvious that 
a heated body tends to grow cold. I am not now 
speaking of fires or of actual combustion whereby 
heat is produced ; I am speaking merely of such 
heat as would be possessed by a red-hot poker 
after being taken from the fire, or by an iron casting 
after the metal has been run into the mould. In 
such cases as this the general law holds good, 
that the heated body tends to grow cold. The 
cooling may be retarded no doubt if the passage 
of heat from the body is impeded. We can, for 
instance, retard the cooling of a teapot by the 
well-known practice of putting a cosy upon it ; but 
the law remains that, slowly or quickly, the heated 
body will tend to grow colder. It seems almost 
puerile to insist with any emphasis 'on a point so 
obvious as this, but yet I frequently find that 
people do not readily apprehend all the gigantic 
consequences that can flow from a principle so 
simple. It is true that a poker cools when taken 
from the fire ; we also find that a gigantic casting 
weighing many tons will grow gradually cold, 
though it may require days to do so. The same 
principle will extend to any object, no matter how 


vast it may happen to be. Were that great casting 
2000 miles in diameter, or were it 8000 miles in 
diameter, it will still steadily part with its heat, 
though no doubt the process of cooling becomes 
greatly prolonged with an increase in the dimen- 
sions of the heated body. The earth and the moon 
cannot escape from the application of these simple 

Let us first speak of the earth. There are 
multitudes of volcanoes in action at the present 
moment in various countries upon this earth. Now 
whatever explanation may be given of the ap- 
proximate cause of the volcanic phenomena, there 
can be no doubt that they indicate the exist- 
ence of heat in the interior of the earth. It may 
possibly be, as some have urged, that the volcanoes 
are^ merely vents for comparatively small masses 
of subterranean molten matter ; it may be, as 
others more reasonably, in my opinion, believe, that 
the whole interior of the earth is at the tempera- 
ture of incandescence, and that the eruptions of 
volcanoes and the shocks of earthquakes are merely 
consequences of the gradual shrinkage of the 
external crust, as it continually strives to accom- 


modate itself to the lessening bulk of the fluid 
interior. But whichever view we may adopt, it is 
at least obvious that the earth is in part, at all 
events, a heated body, and that the heat is not in 
the nature of a combustion, generated and sustained 
by the progress of chemical action. No doubt there 
may be local phenomena of this description, but 
by far the larger proportion of the earth's internal 
heat seems merely the fervour of incandescence. It 
is to be likened to the heat of the molten iron 
which has been run into the sand, rather than to 
the glowing coals in the furnace in which that iron 
has been smelted. 

There is one volcanic outbreak of such excep- 
tional interest in these modern times that I cannot 
refrain from alluding to it. Doubtless every one 
has heard of that marvellous eruption of Krakatoa, 
which occurred on August 26th and 27th, 1883, 
and gives a unique chapter in the history of 
volcanic phenomena. Not alone was the eruption 
of Krakatoa alarming in its more ordinary mani- 
festations, but it was unparalleled both in the 
vehemence of the shock and in the distance to 
which the effects of the great eruption were propa- 


gated. I speak not now of the great waves of ocean 
that inundated the coasts of Sumatra and Java, 
and swept away thirty-six thousand people, nor do 
I allude to the intense darkness which spread for 
one hundred and eighty miles or more all round. 
I shall just mention the three most important 
phenomena, which demonstrate the energy which 
still resides in the interior of our earth. Place a 
terrestrial globe before you, and fix your attention 
on the Straits of Sunda ; think also of the great 
atmospheric ocean some two or three hundred 
miles deep which envelopes our earth. When a 
pebble is tossed into a pond a beautiful series of 
concentric ripples diverge from it ; so when Kra- 
katoa burst up in that mighty catastrophe, a series 
of gigantic waves were propagated through the 
air ; they embraced the whole globe, converged to 
the antipodes of Krakatoa, thence again diverged, 
and returned to the seat of the volcano ; a second 
time the mighty scries of atmospheric ripples 
spread to the antipodes, and a second time returned. 
Seven times did that scries of waves course over 
our globe, and leave their traces on every self- 
recording barometer that our earth possesses. 


Thirty-six hours were occupied in the journey of 
the great undulation from Krakatoa to its anti- 
podes. Perhaps even more striking was the extent 
of our earth's surface over which the noise of the 
great explosion spread. At Batavia, ninety-four 
miles away, the concussions were simply deafen- 
ing ; at Macassar, in Celebes, two steamers were 
sent out to investigate the ex-plosions which were 
heard, little thinking that they came from Kra- 
katoa, nine hundred and sixty-nine miles away. 
Alarming sounds were heard over the island of 
Timor, one thousand three hundred and fifty-one 
miles away from Krakatoa. Diego Garcia in the 
Chogos islands is two thousand two hundred and 
sixty-seven miles from Krakatoa, but the thunders 
traversed even this distance, and were attributed 
to some ship in distress, for which a search was 
made. Most astounding of all, there is undoubted 
evidence that the sound of the mighty explosion 
was propagated across nearly the entire Indian 
ocean, and was heard in the island of Rodriguez, 
almost three thousand miles away. The im- 
mense distance over which this sound journeyed 
will be appreciated by the fact, that the noise did 

F 2 


not reach Rodriguez until four hours after it had 
left Krakatoa. In fact, it would seem that if 
Vesuvius were to explode with the same vehe- 
mence as Krakatoa did, the thunders of the 
explosion might penetrate so far as to be heard 
in London. 

There is another and more beautiful mani- 
festation of the world-wide significance of the 
Krakatoa outbreak. The vast column of smoke 
and ashes ascended twenty miles high in the 
air, and commenced a series of voyages around 
the equatorial regions of the earth. In three days 
it crossed the Indian ocean, and was traversing 
equatorial Africa ; then came an Atlantic voyage ; 
and then it coursed over central America, before 
a Pacific voyage brought it back to its point of 
departure after thirteen days ; then the dust started 
again, and was traced around another similar 
circuit, while it was even tracked for a considerable 
time in placing the third girdle round the earth. 
Strange blue suns and green moons and other 
mysterious phenomena marked the progress of this 
vast volcanic cloud. At last the cloud began to 
lose its density, the dust spread more widely 


over the tropics, became diffused through the 
temperate regions, and then the whole earth was 
able to participate in the glories of Krakatoa. 
The marvellous sunsets in the autumn of 1883 
are attributable to this cause ; and thus once again 
was brought before us the fact that the earth 
still contains large stores of thermal energy. 

Attempts are sometimes made to explain volcanic 
phenomena on the supposition that they are 
entirely of a local character, and that we are not 
entitled to infer the incandescent nature of the 
earth's interior from the fact that volcanic out- 
breaks occasionally happen. For our present pur- 
pose this point is immaterial, though I must say 
it appears to me unreasonable to deny that the 
interior of the earth is in a most highly heated 
state. Every test we can apply shows us the 
existence of internal heat. Setting aside the more 
colossal phenomena of volcanic eruptions, we have 
innumerable minor manifestations of its presence. 
Are there not geysers and hot springs in many 
parts of the earth ? and have we not all over our 
globe invariable testimony confirming the state- 
ment, that the deeper we go down beneath its 


surface the hotter does the temperature become ? 
Every miner is familiar with these facts ; he knows 
that the deeper are his shafts the warmer it is 
down below, and the greater the necessity for pro- 
viding increased ventilation to keep the tempera- 
ture within a limit that shall be suitable for the 
workmen. All these varied classes of phenomena 
admit solely of one explanation, and that is, that 
the interior of the earth contains vast stores of 
incandescent heat. 

We now apply to our earth the same reasoning 
which we should employ on a poker taken from the 
fire, or on a casting drawn from the foundry. Such 
bodies will lose their heat by radiation and conduc- 
tion. The earth is therefore losing its heat. No doubt 
the process is an extremely slow one. The mighty 
reservoirs of internal heat are covered by vast 
layers of rock, which are such excellent non-con- 
ductors that they offer every possible impediment 
to the leakage of heat from the interior to the 
surface. We coat our steam-pipes over with non- 
conducting material, and this can now be done so 
successfully, that it is beginning to be found 
economical to transmit steam for a very long 


distance through properly protected pipes. But 
no non-conducting material that we can manu- 
facture can be half so effective as the shell of rock 
twenty miles or more in thickness, which secures 
the heated interior of the earth from rapid loss 
by radiation into space. Even were the earth's 
surface solid copper or solid silver, both most 
admirable conductors of heat, the cooling down 
of this vast globe would be an extremely tardy 
process ; how much more tardy must it therefore 
be when such exceedingly bad conductors as rocks 
form the envelope ? How imperfectly material 
of this kind will transmit heat is strikingly illus- 
trated by the great blast iron furnaces which are 
so vitally important in one of England's greatest 
manufacturing industries. A glowing mass of coal 
and iron ore and limestone is here urged to vivid 
incandescence by a blast of air itself heated 
to an intense temperature. The mighty heat thus, 
generated sufficient as it is to detach the iron 
from its close alliance with the earthy materials 
and to render the metal out as a pure stream 
rushing white-hot from the vent is sufficiently 
confined by a few feet of brick-work, one side of 


which is therefore at the temperature of molten 
iron, while the other is at a temperature not much 
exceeding that of the air. We may liken the brick- 
work of a blast furnace to the rocky covering of 
the earth ; in each case an exceedingly high tem- 
perature on one side is compatible with a very 
moderate temperature on the other. 

Although the drainage of heat away from the 
earth's interior to its surface, and its loss there by 
radiation into space, is an extremely tardy process, 
yet it is incessantly going on. We have here 
again to note the ability for gigantic effect which 
a small but continually operating cause may have, 
provided it always tends in the same direction. 
The earth is incessantly losing heat ; and though 
in a day, a week, or a year the loss may not be 
very significant, yet when we come to deal with 
periods of time that have to be reckoned by millions 
of years, it may well be that the effect of a small 
loss of heat per annum can, in the course of these 
ages, reach unimagined dimensions. Suppose, for 
instance, that the earth experienced a fall of tem- 
perature in its interior which amounted to only one- 
thousandth of a degree in a year. So minute a 


quantity as this is imperceptible. Even in a century, 
the loss of heat at this rate would be only the tenth 
of a degree. There would be no possible way of 
detecting it ; the most careful thermometer could 
not be relied on to tell us for a certainty that the 
temperature of the hot waters of Bath had declined 
the tenth of a degree ; and I need hardly say, that 
the fall of a tenth of a degree would signify nothing 
in the lavas of Vesuvius, nor influence the thunders 
of Krakatoa by one appreciable note. So far 
as a human life or the life of the human race 
is concerned, the decline of a tenth of a degree 
per century in the earth's internal heat is abso- 
lutely void of significance. I cannot, however, 
impress upon you too strongly, that the mere few 
thousands of years with which human history is 
cognizant are an inappreciable moment in com- 
parison with those unmeasured millions of years 
which geology opens out to us, or with those far 
more majestic periods which the astronomer de- 
mands for the events he has to describe. 

An annual loss of even one-thousandth of a degree 
will be capable of stupendous achievements when 
supposed to operate during epochs of geological 


magnitude. In fact, its effects would be so vast, 
that it seems hardly credible that the present loss 
of heat from the earth should be so great as to 
amount to an abatement of one-thousandth of a 
degree per annum, for that would mean, that in 
a thousand years the earth's temperature would 
decline by one degree, and in a million years the 
decline would amount to a thousand degrees. At 
all events, the illustration may suffice to show, that 
the fact that we are not able to prove by our instru- 
ments that the earth is cooling is no argument 
whatever against the inevitable law, that the earth, 
like every other heated body, must be tending 
towards a lower temperature. 

Without pretending to any numerical accuracy, 
we can at all events give a qualitative if not a 
quantative analysis of the past history of our earth, 
in so far as its changes of temperature are concerned. 
A million years ago our earth doubtless contained 
appreciably more heat than it does at present. I 
speak not, of course, of mere solar heat that is, 
of the heat which now maintains organic life ; I 
am only referring to the original hoard of internal 
heat which is gradually waning. As therefore our 


retrospect extends through millions and millions 
of past ages, we see our earth ever growing warmer 
and warmer the further and further we look back. 
There was a time when those high temperatures 
which we have now found deep down in mines 
were considerably nearer the surface. At present, 
were it not for the sun, the heat of the earth where 
we stand would hardly be appreciably above the 
temperature of infinite space perhaps some 300 
degrees or more below zero. But there must have 
been a time when there was sufficient internal heat 
to maintain the exterior at a warm and indeed at 
a very hot temperature. Nor, so far as we know, 
is there any bound to our retrospect arising from 
the operation or intervention of any other agent. 
The further and the further we look back, the 
hotter and the hotter grows the earth. Nor can we 
stop until, at an antiquity so great that I do not 
venture on any estimate of the date, we discover 
that even at its surface this earth must have con- 
sisted of glowing hot material. We can look back 
still earlier, and see the rocks or whatever other 
term we choose to apply to the then ingredients of 
the earth's crust in a white-hot and even in a 


molten condition. Thus our argument has led us 
to the belief that time was when this now solid 
globe of ours was a ball of white-hot fluid. 

On the argument which I have here used there 
are just two remarks which I particularly wish to 
make. Note in the first place, that our reasoning 
is founded on the fact that the earth is at present, 
to some extent, heated. It matters not whether 
this heat be much or little ; our argument would 
have been equally valid had the earth only con- 
tained a single particle of its mass at a some- 
what higher temperature than the temperature 
of space. I am, of course, not alluding in this to 
heat which can be generated by combustion. The 
other point to which I refer is to remove an ob- 
jection which may possibly be urged against this 
line of reasoning. I have argued that because the 
temperature is continually increasing as we look 
backwards, that therefore a very great temperature 
must once have prevailed. Without some explan- 
ation this argument is not logically complete. 
There is, it is well known, the old paradox of the 
geometric series ; you may add a farthing to a 
halfpenny, and then a half-farthing, and then a 


quarter-farthing, and then the eighth of a farthing, 
followed by the sixteenth, and thirty-second, and 
so on, halving the contribution each time. Now 
no matter how long you continue this process, even 
if you went on with it for ever, and thus made an 
infinite number of contributions, you would never 
accomplish the task of raising the original half- 
penny to the dignity of a penny. An infinite 
number of quantities may therefore, as this illus- 
tration shows, never succeed in attaining any con- 
siderable dimensions. Our argument, however, 
with regard to the increase of heat as we look 
back is the very opposite of this. It is the essence 
of a cooling body to lose heat more rapidly in 
proportion as its temperature is greater. Thus 
though the one-thousandth of a degree may be all 
the fall of temperature that our earth now experi- 
ences in a twelvemonth, yet in those glowing days 
when the surface was heated to incandescence, the 
loss of heat per annum must have been immensely 
greater than it is now. It therefore follows that 
the rate of gain of the earth's heat as we look back 
must be of quite a different character to that of 
the geometric series which I have just illustrated; 


for each addition to the earth's heat, as we look 
back from year to year, must grow greater and 
greater, and therefore there is here no shelter 
for a fallacy in the argument on which the exist- 
ence of high temperature of primeval times is 

The reasoning that I have applied to our earth 
may be applied in almost similar words to the 
moon. It is true that we have not any knowledge 
of the internal nature of the moon at present, nor 
are we able to point to any active volcanic 
phenomena at present in progress there in sup- 
port of the contention that the moon either has 
now internal heat, or did once possess it. It is, 
however, impossible to deny the evidence which 
the lunar craters afford as to the past existence 
of volcanic activity on our satellite. Heat, there- 
fore, there was once in the moon ; and accordingly 
we are enabled to conclude that, on a retrospect 
through illimitable periods of time, we must find 
the moon transformed from that cold and inert 
body she now seems to a glowing and incandes- 
cent mass of molten material. The earth therefore 
and the moon in some remote ages not alone 


anterior to the existence of life, but anterior even 
to the earliest periods of which geologists have 
cognizance must have been both globes of molten 
materials which have consolidated into the rocks 
of the present epoch. 

We must now revert to the tidal history of the 
earth-moon system. Did we not show that there 
was a time when the earth and the moon or 
perhaps, I should say, the ingredients of the earth 
and moon were close together, were indeed in 
actual contact? We have now learned, from a 
wholly different line of reasoning, that in very 
early ages both bodies were highly heated. Here 
as elsewhere in this theory we can make little or 
no attempt to give any chronology, or to har- 
monize the different lines along which the course 
of history has run. No one can form the slightest 
idea as to what the temperature of the earth and of 
the moon must have been in those primeval ages 
when they were in contact. It is impossible, how- 
ever, to deny that they must both have been in 
a very highly heated state ; and everything we 
know of the matter inclines us to the belief that 
the temperature of the earth-moon system must 


at this critical epoch have been one of glowing 
incandescence and fusion. It is therefore quite pos- 
sible that these bodies the moon especially may 
not then have possessed the form in which we now 
find them. It has been supposed, and there are 
some grounds for the supposition, that at this initial 
stage of earth-moon history the moon materials 
did not form a globe, but were disposed in a ring 
which surrounded the earth, the ring being in a 
condition of rapid rotation. It was at a subsequent 
period, according to this view, that the substances 
in the ring gradually drew together, and then by 
their mutual attractions formed a globe which ulti- 
mately consolidated down into the compact moon 
as we now see it. I must, however, specially draw 
your attention to the clearly-marked line which 
divides the facts which dynamics have taught us 
from those notions which are to be regarded as 
more or less conjectural. Interpreting the action 
of the tides by the principles of dynamics, we are 
assured that the moon was once or rather the 
materials of the moon in the immediate vicinity 
of the earth. There, however, dynamics leaves us, 
and unfortunately withholds its accurate illumin- 


ation from the events which immediately preceded 
that state of things. 

The theory of tidal evolution which I am de- 
scribing in these lectures is mainly the work of 
Professor George H. Darwin of Cambridge. Much 
of the original parts of the theory of the tides was 
due to Sir William Thomson, and I have also 
mentioned how Professor Purser contributed an 
important element to the dynamical theory. It 
is, however, Darwin who has persistently deduced 
from the theory the various consequences which 
can be legitimately drawn from it. Darwin, for 
instance, pointed out that as the moon is receding 
from us, it must, if we only look far enough back, 
have been once in practical contact with the earth. 
It is to Darwin also that we owe many of the 
other parts of a fascinating theory, either in its 
mathematical or astronomical aspect ; but I must 
take this opportunity of saying, that I do not 
propose to make Professor Darwin or any of the 
other mathematicians I have named responsible 
for all that I shall say in these lectures. I must 
be myself accountable for the way in which the 
subject is being treated, as well as for many of 



the illustrations used, and some of the deductions 
I have drawn from the subject. 

It is almost unavoidable for us to make a sur- 
mise as to the cause by which the moon had come 
into this remarkable position close to the earth at 
the most critical epoch of earth-moon history. 

With reference to this Professor Darwin has 
offered an explanation, which seems so exceedingly 
plausible that it is impossible to resist the notion 
that it must be correct. I will ask you to think 
of the earth not as a solid body covered largely 
with ocean, but as a glowing globe of molten 
material. In a globe of this kind it is possible for 
great undulations to be set up. Here is a large vase 
of water, and by displacing it I can cause the water 
to undulate with a period which depends on the 
size of the vessel ; undulations can be set up in a 
bucket of water, the period of these undulations 
being dependent upon the dimensions of the bucket. 
Similarly in a vast globe of molten material certain 
undulations could be set up, and those undulations 
would have a period depending upon the dimen- 
sions of this vibrating mass. We may conjecture a 
mode in which such vibrations could be originated. 


Imagine a thin shell of rigid material which just 
encases the globe; suppose this be divided into 
four quarters, like the four quarters of an orange, 
and that two of these opposite quarters be rejected, 
leaving two quarters on the liquid. Now suppose 
that these two quarters be suddenly pressed in, and 
then be as suddenly removed they will produce 
depressions, of course, on the two opposite quarters, 
while the uncompressed quarters will become 
protuberant. In virtue of the mutual attractions 
between the different particles of the mass, an 
effort will be made to restore the globular form, 
but this will of course rather overshoot the mark ; 
and therefore a series of undulations will be origin- 
ated by which two opposite quarters of the sphere 
will alternately shrink in and become protuberant. 
There will be a particular period to this oscillation. 
For our globe it would appear to be somewhere 
about an hour and a half or two hours ; but there 
is necessarily a good deal of uncertainty about this 

We have seen how in those primitive days the 
earth was spinning around very rapidly ; and I have 
also stated that the earth might at this very 

G 2 


critical epoch of its history be compared with a 
grindstone which is being driven so rapidly that 
it is on the very brink of rupture. It is remark- 
able to note, that a cause tending to precipitate 
a rupture of the earth was at hand. The sun then 
raised tides in the earth as it does at present. 
When the earth revolved in a period of some four 
hours or thereabouts, the high tides caused by the 
sun succeeded each other at intervals of about two 
hours. When I speak of tides in this respect, 
of course I am not alluding to oceanic tides ; 
these were the days long before ocean existed, at 
least in the liquid form. The tides I am speaking 
about were raised in the fluids and materials which 
then constituted the whole of the glowing earth ; 
those tides rose and fell under the throb produced 
by the sun, just as truly as tides produced in an 
ordinary ocean. But now note the significant coin- 
cidence between the period of the throb produced 
by the sun-raised tides, and that natural period of 
vibration which belonged to our earth as a mass 
of molten material. It therefore follows, that the 
impulse given to the earth by the sun harmonized 
in time with that period in which the earth itself 


was disposed to oscillate. A well-known dynamical 
principle here comes into play. You see a heavy 
weight hanging by a string, and in my hand I 
hold a little slip of wood no heavier than a com- 
mon pencil ; ordinarily speaking, I might strike 
that heavy weight with this slip of wood, and no 
effect is produced ; but if I take care to time the 
little blows that I give so that they shall harmonize 
with the vibrations which the weight is naturally 
disposed to make, then the effect of many small 
blows will be cumulative, so much so, that after a 
short time the weight begins to respond to my 
efforts, and now you see it has acquired a swing of 
very considerable amplitude. 

Among the remarkable discoveries which have 
been made in modern times are those astound- 
ing experiments of Hertz, in which well-timed 
electrical impulses broke down an air resistance, 
and revealed to us ethereal vibrations which could 
never have been made manifest except by the 
principle we are here discussing. The ingenious 
conjecture has been made, that when the earth was 
thrown into tidal vibrations in those primeval days, 
these slight vibrations, harmonizing as they did 


with the natural period of the earth, gradually 
acquired amplitude ; the result being that the pulse 
of each successive vibration increased at last to 
such an extent that the earth separated under the 
stress, and threw off a portion of those semi-fluid 
materials of which it was composed. In process of 
time these rejected portions contracted together, 
and ultimately formed that moon we now see. 
Such is the origin of the moon which the modern 
theory of tidal evolution has presented to our 

There are two great epochs in the evolution of 
the earth-moon system two critical epochs which 
possess a unique dynamical significance ; one of 
these periods was early in the beginning, while the 
other cannot arrive for countless ages yet to come. 
I am aware that in discussing this matter I am 
entering somewhat largely into mathematical prin- 
ciples ; I must only endeavour to state the matter 
as succinctly as the subject will admit. 

In an earlier part of this lecture I have ex- 
plained how, during all the development of the 
earth-moon system, the quantity of moment of 
momentum remains unaltered. The moment of 


momentum of the earth's rotation added to the 
moment of momentum of the moon's revolution 
remains constant ; if one of these quantities in- 
crease the other must decrease, and the progress of 
the evolution will have this result, that energy shall 
be gradually lost in consequence of the friction 
produced by the tides. The investigation is one 
appropriate for mathematical formulae, such as 
those that can be found in Professor Darwin's 
memoirs ; but nature has in this instance dealt 
kindly with us, for she has enabled an abstruse 
mathematical principle to be dealt with in a singu- 
larly clear and concise manner. We want to obtain 
a definite view of the alteration in the energy of 
the system which shall correspond to a small 
change in the velocity of the earth's rotation, the 
moon of course accommodating itself so that the 
moment of momentum shall be preserved un- 
altered. We can use for this purpose an angular 
velocity which represents the excess of the earth's 
rotation over the angular revolution of the moon ; 
it is, in fact, the apparent angular velocity with 
which the moon appears to move round the 
heavens. If we represent by N the angular 


velocity of the earth, and by M the angular 
velocity of the moon in its orbit round the earth, 
the quantity we desire to express is N M ; 
we shall call it the relative rotation. The mathe- 
matical theorem which tells us what we want can 
be enunciated in a concise manner as follows. 
The alteration of the energy of the system may be 
expressed by multiplying the relative rotation by the 
change in the earth's angular velocity. This result 
will explain many points to us in the theory, but 
just at present I am only going to make a single 
inference from it. 

I must advert for a moment to the familiar 
conception of a maximum or a minimum. If a 
magnitude be increasing, that is, gradually growing 
greater and greater, it has obviously not attained 
a maximum so long as the growth is in progress. 
Nor if the object be actually decreasing can it be 
said to be at a maximum either ; for then it was 
greater a second ago than it is now, and therefore 
it cannot be at a maximum at present. We may 
illustrate this by the familiar example of a stone 
thrown up into the air; at first it gradually rises, 
being higher at each instant than it was previously, 


until a culminating point is reached, when just for 
a moment the stone is poised at the summit of its 
path ere it commences its return to earth again. 
In this case the maximum point is obtained when 
the stone, having ceased to ascend, and not having 
yet commenced to descend, is momentarily at rest. 

The same principles apply to the determination 
of a minimum. As long as the magnitude is de- 
clining the minimum has not been reached ; it is 
only vvhen the decline has ceased, and an increase 
is on the point of setting in, that the minimum 
can be said to be touched. 

The earth-moon system contains at any moment 
a certain store of energy, and to every conceivable 
condition of the earth-moon system a certain 
quantity of energy is appropriate. It is instructive 
for us to study the different positions in which 
the earth and the moon might lie, and to examine 
the different quantities of energy which the system 
will contain in each of those varied positions. It 
is however to be understood that the different 
cases all presuppose the same total moment of 

Among the different cases that can be imagined, 


those will be of special interest in which the total 
quantity of energy in the system is a maximum 
or a minimum. We must for this purpose suppose 
the system gradually to run through all conceiv- 
able changes, with the earth and moon as near as 
possible, and as far as possible, and in all inter- 
mediate positions ; we must also attribute to the 
earth every variety in the velocity of its rotation 
which is compatible with the preservation of the 
moment of momentum. Beginning then with the 
earth's velocity of rotation at its lowest, we may 
suppose it gradually and continually increased, and, 
as we have already seen, the change in the energy 
of the system is to be expressed by multiplying 
the relative rotation into the change of the earth's 
angular velocity. It follows from the principles 
we have already explained, that the maximum 
or minimum energy is attained at the moment 
when the alteration is zero. It therefore follows, 
that the critical periods of the system will arise 
when the relative rotation is zero, that is, when 
the earth's rotation on its axis is performed with 
a velocity equal to that with which the moon 
revolves around the earth. This is truly a singular 


condition of the earth-moon system ; the moon in 
such a case would revolve around the earth as 
if the two bodies were bound together by rigid 
bonds into what was practically a single solid 
body. At the present moment no doubt to some 
extent this condition is realized, because the moon 
always turns the same face to the earth (a point on 
which we shall have something to say later on) ; but 
in the original condition of the earth-moon system, 
the earth would also constantly direct the same 
face to the moon, a condition of things which is 
now very far from being realized. 

It can be shown from the mathematical nature 
of the problem that there are four states of the 
earth-moon system in which this condition may 
be realized, and which are also compatible with 
the conservation of the moment of momentum. 
We can express what this condition implies in a 
somewhat more simple manner. Let us under- 
stand by the day the period of the earth's rotation 
on its axis, whatever that may be, and let us 
understand by the month the period of revolution 
of the moon around the earth, whatever value it 
may have; then the condition of maximum or 


minimum energy is attained when the day and the 
month have become equal to each other. Of the 
four occasions mathematically possible in which 
the day and the month can be equal, there are 
only two which at present need engage our at- 
tention one of these occurred near the beginning 
of the earth and moon's history, the other remains 
to be approached in the immeasurably remote 
future. The two remaining solutions are futile, 
being what the mathematician would describe as 

There is a fundamental difference between the 
dynamical conditions in these critical epochs in 
one of them the energy of the system has attained 
a maximum value, and in the other the energy of 
the system is at a minimum value. It is impos- 
sible to over-estimate the significance of these two 
states of the system. 

I may recall the fundamental notion which every 
one has learned in mechanics, as to the difference 
between stable and unstable equilibrium. The 
conceivable possibility of making an egg stand on 
its end is a practical impossibility, because nature 
does not like unstable equilibrium, and a body 


departs therefrom on the least disturbance ; on 
the other hand, stable equilibrium is the position in 
which nature tends to place everything. A log of 
wood floating on a river might conceivably float 
in a vertical position with its end up out of the 
water, but you never could succeed in so balancing 
it, because no matter how carefully you adjusted 
the log, it would almost instantly turn over when 
you left it free ; on the other hand, when the log 
floats naturally on the water it assumes a hori- 
zontal position, to which, when momentarily dis- 
placed therefrom, it will return if permitted to 
do so. We have here an illustration of the con- 
trast between stable and unstable equilibrium. 
It will be found generally that a body is in 
equilibrium when its centre of gravity is at its 
highest point or at its lowest point ; there is, 
however, this important difference, that when the 
centre of gravity is highest the equilibrium is 
unstable, and when the centre of gravity is lowest 
the equilibrium is stable. The potential energy 
of an egg poised on its end in unstable equi- 
librium is greater than when it lies on its side 
in stable equilibrium. In fact, energy must be 


expended to raise the egg from the horizontal 
position to the vertical ; while, on the other hand, 
work could conceivably be done by the egg when 
it passes from the vertical position to the horizontal. 
Speaking generally, we may say that the stable 
position indicates low energy, while a redundancy 
of that valuable agent is suggestive of instability. 

We may apply similar principles to the con- 
sideration of the earth-moon system. It is true 
that we have here a series of dynamical phe- 
nomena, while the illustrations I have given of 
stable and unstable equilibrium relate only to 
statical problems ; but we can have dynamical sta- 
bility and dynamical instability, just as we can 
have stable and unstable equilibrium. Dynamical 
instability corresponds with the maximum of 
energy, and dynamical stability to the minimum 
of energy. 

At that primitive epoch, when the energy of the 
earth-moon system was a maximum, the condition 
was one of dynamical instability ; it was impossible 
that it should last. But now mark how truly 
critical an occurrence this must have been in the 
history of the earth-moon system, for have I not 


already explained that it is a necessary condition 
of the progress of tidal evolution that the energy 
of the system should be always declining ? But 
here our retrospect has conducted us back to a 
most eventful crisis, in which the energy was a 
maximum, and therefore cannot have been imme- 
diately preceded by a state in which the energy 
was greater still ; it is therefore impossible for the 
tidal evolution to have produced this state of 
things ; some other influence must have been in 
operation at this beginning of the earth-moon 

Thus there can hardly be a doubt that imme- 
diately preceding the critical epoch the moon 
originated from the earth in the way we have 
described. Note also that this condition, being 
one of maximum energy, was necessarily of dy- 
namical instability, it could not last ; the moon 
must adopt either of two courses it must tumble 
back on the earth, or it must start outwards. 
Now which course was the moon to adopt ? The 
case is analogous to that of an egg standing on 
its end it will inevitably tumble one way or 
the other. Some infinitesimal cause will produce 


a tendency towards one side, and to that side 
accordingly the egg will fall. The earth-moon 
system was similarly in an unstable state, an in- 
finitesimal cause might conceivably decide the fate 
of the system. We are at present in ignorance 
of what the determining cause might have been, 
but the effect it produced is perfectly clear; the 
moon did not again return to its mother earth, but 
set out on that mighty career which is in progress 

Let it be noted that these critical epochs in 
the earth - rnoon history arise when and only 
when there is an absolute identity between the 
length of the month and the length of the day. 
It may be proper therefore that I should pro- 
vide a demonstration of the fact, that the identity 
between these two periods must necessarily 
have occurred at a very early period in the 

The law of Kepler, which asserts that the square 
of the periodic time is proportioned to the cube 
of the mean distance, is in its ordinary application 
confined to a comparison between the revolutions 
of the several planets about the sun. The periodic 


time of each planet is connected with its average 
distance by this law ; but there is another appli- 
cation of Kepler's law which gives us information 
of the distance and the period of the moon in 
former stages of the earth-moon history. Al- 
though the actual path of the moon is of course 
an ellipse, yet that ellipse is troubled, as is well 
known, by many disturbing forces, and from this 
cause alone the actual path of the moon is far 
from being any of those simple curves with which 
we are so well acquainted. Even were the earth 
and the moon absolutely rigid particles, perturb- 
ations would work all sorts of small changes in 
the pliant curve. The phenomena of tidal evolution 
impart an additional element of complexity into 
the actual shape of the moon's path. We now see 
that the ellipse is not merely subject to incessant 
deflections of a periodic nature, it also undergoes 
a gradual contraction as we look back through 
time past ; but we may, with all needful accuracy 
for our present purpose, think of the path of the 
moon as a circle, only we must attribute to that 
circle a continuous contraction of its radius the 
further and the further we look back. The alteration 



in the radius will be even so slow, that the moon 
will accomplish thousands of revolutions around 
the earth without any appreciable alteration in 
the average distance of the two bodies. We can 
therefore think of the moon as revolving at every 
epoch in a circle of special radius, and as accom- 
plishing that revolution in a special time. With 
this understanding we can now apply Kepler's law 
to the several stages of the moon's past history. 
The periodic time of each revolution, and the mean 
distance at which that revolution was performed, 
will be always connected together by the formula 
of Kepler. Thus to take an instance in the very 
remote past. Let us suppose that the moon was 
at one hundred and twenty thousand miles instead 
of two hundred and forty thousand, that is, at 
half its present distance. Applying the law of 
Kepler, we see that the time of revolution must 
then have been only about ten days instead of the 
twenty-seven it is now. Still further, let us sup- 
pose that the moon revolves in an orbit with one- 
tenth of the diameter it has at present, then the 
cube of 10 being 1000, and the square root of 1000 
being 31 '6, it follows that the month must have 


been less than the thirty-first part of what it is 
at present, that is, it must have been considerably 
less than one of our present days. Thus you see the 
month is growing shorter and shorter the further 
we look back, the day is also growing shorter and 
shorter ; but still I think we can show that there 
must have been a time when the month will have 
been at least as short as the day. 

Consider the case when the moon shall have 
made the closest possible approximation to the 
earth. Two globes in contact will have a distance 
between their centres which is equal to the sum 
of their radii. Take the earth as having a radius 
of four thousand miles, and the moon a radius 
of one thousand miles, the two centres must 
at their shortest distance be five thousand miles 
apart, that is, the moon must then be at the forty- 
eighth part of its present distance from the earth. 
Now the cube of 48 is 110,592, and the square root 
of 110,592 is nearly 333, therefore the length of the 
month will be one-three hundred and thirty-third 
part of the duration of the month at present ; in 
other words, the moon must revolve around the 

earth in a period of somewhat about two hours. It 

II 2 


seems impossible that the day can ever have been 
as brief as this. We have therefore proved that, in 
the course of its contracting duration, the moon 
must have overtaken the contracting day, and that 
therefore there must have been a time when the 
moon was in the vicinity of the earth, and having 
a day and month of equal period. Thus we have 
shown that the critical condition of dynamical 
instability must have occurred in the early period 
of the earth-moon history, if the agents then in 
operation were those which we now know. The 
further development of the subject must be post- 
poned until the next lecture. 



STARTING from that fitting commencement of 
earth-moon history which the critical epoch affords, 
we shall now describe the dynamical phenomena 
as the tidal evolution progressed. The moon and 
the earth initially moved as a solid body, each 
bending the same face towards the other ; but as 
the moon retreated, and as tides began to be 
raised on the earth, the length of the day began 
to increase, as did also the length of the month. 
We know, however, that the month increased more 
rapidly than the day, so that a time was reached 
when the month was twice as long as the day ; 
and still both periods kept on increasing, but not 
at equal rates, for in progress of time the month 
grew so much more rapidly than the day, that 
many days had to elapse while the moon accom- 
plished a single revolution. It is, however, only 
necessary for us to note those stages of the mighty 


progress which correspond to special events. The 
first of such stages was attained when the month 
assumed its maximum ratio to the day. At this 
time, the month was about twenty-nine days, 
and the epoch appears to have occurred at a 
comparatively recent date if we use such 
standards of time as tidal evolution requires ; 
though measured by historical standards, the epoch 
is of incalculable antiquity. I cannot impress 
upon you too often the enormous magnitude of 
the period of time which these phenomena have 
required for their evolution. Professor Darwin's 
theory affords but little information on this point, 
and the utmost we can do is to assign a minor 
limit to the period through which tidal evolution 
has been in progress. It is certain that the birth 
of the moon must have occurred at least fifty 
million years ago, but probably the true period 
is enormously greater than this. If indeed we 
choose to add a cipher or two to the figure just 
printed, I do not think there is anything which 
could tell us that we have over-estimated the 
mark. Therefore, when I speak of the epoch in 
which the month possessed the greatest number of 


days as a recent one, it must be understood that 
I am merely speaking of events in relation to the 
order of tidal evolution. Viewed from this stand- 
point, we can show that the epoch is a recent one 
in the following manner. At present the month 
consists of a little more than twenty-seven days, 
but at this maximum period to which I have 
referred the month was about twenty- nine days ; 
from that it began to decline, and the decline 
cannot have proceeded very far, for even still there 
are only two days less in the month than at the 
time when the month had the greatest number of 
days. It thus follows that the present epoch the 
human epoch, as we may call it in the history 
of the earth has fallen at a time when the progress 
of tidal evolution is about half-way between the 
initial and the final stage. I do not mean half- 
way in the sense of actual measurement of years ; 
indeed, from this point it would seem that we 
cannot yet be nearly half-way, for, vast as are the 
periods of time that have elapsed since the moon 
first took its departure from the earth, they fall far 
short of that awful period of time which will inter- 
vene between the present moment and the hour 


when the next critical state of earth-moon history 
shall have been attained. In that state the day is 
destined once again to be equal to the month, just 
as was the case in the initial stage. The half-way 
stage will therefore in one sense be that in which 
the proportion of the month to the day culminates. 
This is the stage which we have but lately passed ; 
and thus it is that at present we may be said to 
be about half-way through the progress of tidal 

My narrative of the earth-moon evolution must 
from this point forward cease to be retrospective. 
Having begun at that critical moment when the 
month and day were first equal, we have traced 
the progress of events to the present hour. What 
we have now to say is therefore in the nature 
of a forecast. So far as we can tell, no agent 
is likely to interfere with the gradual evolution 
caused by the tides, which dynamical principles 
have disclosed to us. As the years roll on, or 
perhaps, I should rather say, as thousands of 
years and millions of years roll on, the day will 
continue to elongate, or the earth to rotate more 
slowly on its axis. But countless ages must 


elapse before another critical stage of the history 
shall be reached. It is needless for me to ponder 
over the tedious process by which this interesting 
epoch is reached. I shall rather sketch what 
the actual condition of our system will be when 
that moment shall have arrived. The day will 
then have expanded from the present familiar 
twenty-four hours up to a day more than twice, 
more than five, even more than fifty times its 
present duration. In round numbers, we may say 
that this great day will occupy one thousand four 
hundred of our ordinary hours. To realize the 
critical nature of the situation then arrived at, we 
must follow the corresponding evolution through 
which the moon passes. From its present distance 
of two hundred and forty thousand miles, the 
moon will describe an ever-enlarging orbit ; and 
as it does so the duration of the month will also 
increase, until at last a point will be reached when 
the month has become more than double its 
present length, and has attained the particular 
value of one thousand four hundred hours. We 
are specially to observe that this one - thousand 
four-hundred-hour month will be exactly reached 


when the day has also expanded to one thousand 
four hundred hours ; and the essence of this critical 
condition, which may be regarded as a significant 
point of tidal evolution, is that the day and the 
month have again become equal. The day and 
the month were equal at the beginning, the day 
and the month will bs equal at the end. Yet how 
wide is the difference between the beginning and 
the end. The day or the month at the end is some 
hundreds of times as long as the month or the day 
at the beginning. 

I have already fully explained how, in any stage 
of the evolutionary progress in which the day and 
the month became equal, the energy of the system 
attained a maximum or a minimum value. At 
the beginning the energy was a maximum ; at the 
end the energy will be a minimum. The most 
important consequences follow from this consider- 
ation. I have already shown that a condition of 
maximum energy corresponded to dynamic insta- 
bility. Thus we saw that the earth-moon history 
could not have commenced without the interven- 
tion of some influence other than tides at the 
beginning. Now let us learn what the similar 


doctrine has to tell us with regard to the end. 
The condition then arrived at is one of dynamical 
stability ; for suppose that the system were to 
receive a slight alteration, by which the moon went 
out a little further, and thus described a larger 
orbit, and so performed more than its share of the 
moment of spin. Then the earth would have to 
do a little less spinning, because, under all circum- 
stances, the total quantity of spin must be pre- 
served unaltered. But the energy being at a 
minimum, such a small displacement must of 
course produce a state of things in which the 
energy would be increased. Or if we conceived 
the moon to come in towards the earth, the moon 
would then contribute less to the total moment 
of momentum. It would therefore be incumbent 
on the earth to do more ; and accordingly the 
velocity of the earth's rotation would be aug- 
mented. But this arrangement also could only be 
produced by the addition of some fresh energy 
to the system, because the position from which 
the system is supposed to have been disturbed 
is one of minimum energy. 

No disturbance of the system from this final 


position is therefore conceivable, unless some 
energy can be communicated to it. But this will 
demonstrate the utter incompetency of the tides 
to shift the system by a hair's breadth from this 
position ; for it is of the essence of the tides to 
waste energy by friction. And the transformations 
of the system which the tides have caused are 
invariably characterized by a decline of energy, 
the movements being otherwise arranged so that 
the total moment of momentum shall be preserved 
intact. Note, how far we were justified in speak- 
ing of this condition as a final one. It is final 
so far as the lunar tides are concerned ; and were 
the system to be screened from all outer inter- 
ference, this accommodation between the earth 
and the moon would be eternal. 

There is indeed another way of demonstrating 
that a condition of the system in which the day 
has assumed equality with the month must neces- 
sarily be one of dynamical equilibrium. We have 
shown that the energy which the tides demand is 
derived not from the mere fact that there are high 
tides and low tides, but from the circumstance 
that these tides do rise and fall ; that in falling and 


rising they do produce currents ; and it is these 
currents which generate the friction by which the 
earth's velocity is slowly abated, its energy wasted, 
and no doubt ultimately dissipated as heat. If 
therefore we can make the ebbing and the flowing 
of the tides to cease, then our argument will disap- 
pear. Thus suppose, for the sake of illustration, 
that at a moment when the tides happened to be 
at high water in the Thames, such a change 
took place in the behaviour of the moon that the 
water always remained full in the Thames, and at 
every other spot on the earth remained fixed at 
the exact height which it possessed at this par- 
ticular moment. There would be no more tidal 
friction, and therefore the system would cease to 
course through that series of changes which the 
existence of tidal friction necessitates. 

But if the tide is always to be full in the Thames, 
then the moon must be always in the same position 
with respect to the meridian, that is, the moon must 
always be fixed in the heavens over London. In 
fact, the moon must then revolve around the earth 
just as fast as London does the month must have 
the same length as the day. The earth must then 


show the same face constantly to the moon, just as 
the moon always does show the same face towards 
the earth ; the two globes will in fact revolve as 
if they were connected with invisible bonds, which 
united them into a single rigid body. 

We need therefore feel no surprise at the ces- 
sation of the progress of tidal evolution when the 
month and the day are equal, for then the move- 
ment of moon-raised tides has ceased. No doubt 
the same may be said of the state at the beginning 
of the history, when the day and the month had 
the brief and equal duration of a few hours. While 
the equality of the two periods lasted there could 
be no tides, and therefore no progress in the 
direction of tidal evolution. There is, however, the 
profound difference of stability and instability be- 
tween the two cases ; the most insignificant dis- 
turbance of the system at the initial stage was 
sufficient to precipitate the revolving moon from its 
condition of dynamical equilibrium, and to start 
the course of tidal evolution in full vigour. If, 
however, any trifling derangement should take 
place in the last condition of the system, so that 
the month and the day departed slightly from 


equality, there would instantly be an ebbing and a 
flowing of the tides ; and the friction generated by 
these tides would operate to restore the equality 
because this condition is one of dynamical stability. 

It will thus be seen with what justice we can 
look forward to the day and month each of four- 
teen hundred hours as a finale to the progress of 
the luni-tidal evolution. Throughout the whole of 
this marvellous series of changes it is always neces- 
sary to remember the one constant and invariable 
element the moment of momentum of the system 
which tides cannot alter. Whatever else the friction 
can have done, however fearful may have been the 
loss of energy by the system, the moment of mo- 
mentum which the system had at the beginning it 
preserves unto the end. This it is which chiefly 
gives us the numerical data on which we have to 
rely for the quantitative features of tidal evolution. 

We have made so many demands in the course 
of these lectures on the capacity of tidal friction 
to accomplish startling phenomena in the evolution 
of the earth-moon system, that it is well for us 
to seek for any evidence that may otherwise be 
obtainable as to the capacity of tides for the 


accomplishment of gigantic operations. I do not 
say that there is any doubt which requires to be 
dispelled by such evidence, for as to the general 
outlines of the doctrine of tidal evolution which 
has been here sketched out there can be no reason- 
able ground for mistrust ; but nevertheless it is 
always desirable to widen our comprehension of 
any natural phenomena by observing collateral 
facts. Now there is one branch of tidal action 
to which I have as yet only in the most incidental 
way referred. We have been speaking of the tides 
in the earth which are made to ebb and flow by 
the action of the moon ; we have now to consider 
the tides in the moon, which are there excited by 
the action of the earth. For between these two 
bodies there is a reciprocity of tidal-making energy 
each of them is competent to raise tides in the 
other. As the moon is so small in comparison with 
the earth, and as the tides on the moon are of 
but little significance in the progress of tidal evo- 
lution, it has been permissible for us to omit them 
from our former discussion. But it is these tides 
on the moon which will afford us a striking illus- 
tration of the competency of tides for stupendous 


tasks. The moon presents a monument to show 
what tides are able to accomplish. 

I must first, however, explain a difficulty which 
is almost sure to suggest itself when we speak of 
tides on the moon. I shall be told that the moon 
contains no water on its surface, and how then, it 
will be said, can tides ebb and flow where there is 
no sea to be disturbed ? There are two answers 
to this difficulty ; it is no doubt true that the moon 
seems at present entirely devoid of water in so far 
as its surface is exposed to us, but it is by no 
means certain that the moon was always in this 
destitute condition. There are very large features 
marked on its map as "seas"; these regions are 
of a darker hue than the rest of the moon's sur- 
face, they are large objects often many hundreds 
of miles in diameter, and they form, in fact, those 
dark patches on the brilliant surface which are 
conspicuous to the unaided eye, and are represented 
in Fig. 3. Viewed in a telescope these so-called 
seas, while clearly possessing no water at the 
present time, are yet widely different from the 
general aspect of the moon's surface. It has 
often been supposed that great oceans once filled 


these basins, and a plausible explanation has 
even been offered as to how the waters they once 

Fig. 3. The Moon. 

contained could have vanished. It has been 
thought that as the mineral substances deep in 


the interior of our satellite assumed the crystalline 
form during the progress of cooling, the demand 
for water of crystallization required for incorpor- 
ation with the minerals was so great that the 
oceans of the moon became entirely absorbed. 
It is, however, unnecessary for our present argument 
that this theory should be correct. Even if there 
never was a drop of water found on our satellite, 
the tides in its molten materials would be quite 
sufficient for our purpose ; anything that tides 
could accomplish would be done more speedily by 
vast tides of flowing lava than by merely oceanic 

There can be no doubt that tides raised on the 
moon by the earth would be greater than the tides 
raised on the earth by the moon. The question is, 
however, not a very simple one, for it depends on 
the masses of both bodies as well as on their 
relative dimensions In so far as the masses are 
concerned, the earth being more than eighty times 
as heavy as the moon, the tides would on this 
account be vastly larger on the moon than on the 
earth. On the other hand, the moon's diameter 
being much less than that of the earth, the effi- 

I 2 


ciency of a tide-producing body in its action on 
the moon would be less than that of the same 
body at the same distance in its action on the 
earth ; but the diminution of the tides from this 
cause would be not so great as their increase from 
the former cause, and therefore the net result 
would be to exhibit much greater tides on the moon 
than on the earth. 

Suppose that the moon had been originally 
endowed with a rapid movement of rotation 
around its axis, the effect of the tides on that 
rotation would tend to check its velocity just in the 
same way as the tides on the earth have effected 
a continual elongation of the day. Only as the 
tides on the moon were so enormously great, 
their capacity to check the moon's speed would 
have corresponding efficacy. As the moon is 
so small a body, it could only offer feeble 
resistance to the unceasing action of the tide, 
and therefore our satellite must succumb to 
whatever the tides desired ages before our earth 
would have been affected to a like extent. It 
must be noticed that the influence of the tidal 
friction is not directed to the total annihilation of 


the rotation of the two bodies affected by it ; the 
velocity is only checked down until it attains such 
a point that the speed in which each body rotates 
upon its axis has become equal to that in which 
it revolves around the tide-producer. The practical 
. effect of such an adjustment is to make the tide- 
agitated body turn a constant face towards its 

I may here note a point about which people 
sometimes find a little difficulty. The moon 
constantly turns the same face towards the 
earth, and therefore people are sometimes apt 
to think that the moon performs no rotation 
whatever around its own axis. But this is indeed 
not the case. The true inference to be drawn 
from the constant face of the moon is, that the 
velocity of rotation about its own axis is equal to 
that of its rotation around the earth ; in fact, the 
moon revolves around the earth in twenty-seven 
days, and its rotation about its axis is performed 
in twenty-seven days also. You may illustrate the 
movement of the moon around the earth by walk- 
ing around a table in a room, keeping all the time 
your face turned towards the table ; in such a case 


as this you not only perform a motion of revolu- 
tion, but you also perform a rotation in an equal 
period. The proof that you do rotate is to be found 
in the fact that during the movement your face is 
being directed successively to all the points of the 
compass. There is no more singular fact in the 
solar system than the constancy of the moon's 
face to the earth. The periods of rotation and 
revolution are both alike ; if one of these periods 
exceeded the other by an amount so small as the 
hundredth part of a second, the moon would in the 
lapse of ages permit us to see that other side which 
is now so jealously concealed. The marvellous 
coincidence between these two periods would be 
absolutely inexplicable, unless we were able to 
assign it to some physical cause. It must be remem- 
bered that in this matter the moon occupies a 
unique position among the heavenly host. The sun 
revolves around on its axis in a period of twenty- 
five or twenty-six days thus we see one side of the 
sun as frequently as we see the other. The side of 
the sun which is turned towards us to-day is almost 
entirely different from that we saw a fortnight ago. 
Nor is the period of the sun's rotation to be iden- 


tified with any other remarkable period in our 
system. If it were equal to the length of the year, 
for instance, or if it were equal to the period of any 
of the other planets, then it could hardly be con- 
tended that the phenomenon as presented by the 
moon was unique ; but the sun's period is not 
simply related, or indeed related at all, to any of 
the other periodic times in the system. Nor do 
we find anything like the moon's constancy of 
face in the behavionr of the other planets. Jupiter 
turns now one face to us and then another. Nor 
is his rotation related to the sun or related to any 
other bDdy, as our moon's motion is related to 
us. It has indeed been thought that in the move- 
ments of the satellites of Jupiter a somewhat 
similar phenomenon may be observed to that in 
the motion of our own satellite. If this be so, the 
causes whereby this phenomenon is produced are 
doubtless identical in the two cases. 

So remarkable a coincidence as that which the 
moon's motion shows could not reasonably be 
explained as a mere fortuitous circumstance ; nor 
need we hesitate to admit that a physical explana- 
tion is required when we find a most satisfactory 


one ready for our acceptance, as was originally 
pointed out by Helmholtz. 

There can be no doubt whatever that the con- 
stancy of the moon's face is the work of ancient 
tides, which have long since ceased to act. We 
have shown that if the moon's rotation had once 
been too rapid to permit of the same face being 
always directed towards us, the tides would operate 
as a check by which the velocity of that rotation 
would be abated. On the other hand, if the moon 
rotated so slowly that its other face would be ex- 
posed to us in the course of the revolution, the tides 
would then be dragged violently over its surface 
in the direction of its rotation ; their tendency 
would thus be to accelerate the speed until the 
angular velocity of rotation was equal to that of 
revolution. Thus the tides would act as a controlling 
agent of the utmost stringency to hurry the moon 
round when it was not turning fast enough, and to 
arrest the motion when going too fast. Peace there 
would be none for the moon until it yielded absolute 
compliance to the tyranny of the tides, and ad- 
justed its period of rotation with exact identity 
to its period of revolution. Doubtless this adjust- 


merit was made countless ages ago, and since that 
period the tides have acted so as to preserve 
the adjustment, as long as any part of the moon 
was in a state sufficiently soft or fluid to respond 
to tidal impression. The present state of the 
moon is a monument to which we may con- 
fidently appeal in support of our contention as 
to the great power of the tides during the ages 
which have passed ; it will serve as an illustration 
of the future which is reserved for our earth in 
ages yet to come, when our globe shall have also 
succumbed to tidal influence. 

It is owing to the smallness of the moon relatively 
to the earth that the tidal process has reached a 
much more advanced stage in the moon than it 
has on the earth ; but the moon is incessant in its 
efforts to bring the earth into the same condition 
which it has itself been forced to assume. Thus 
again we look forward to an epoch in the incon- 
ceivably remote future when tidal thraldom shall be 
supreme, and when the earth shall turn the same 
face to the moon, as the moon now turns the 
same face to the earth. 

In the critical state of things thus looming in 


the dim future, the earth and the moon will con- 
tinue to perform this adjusted revolution in a period 
of about fourteen hundred hours, the two bodies 
being held, as it were, by invisible bands. Such 
an arrangement might be eternal if there were no 
intrusion of tidal influence from any other body ; 
but of course in our system as we actually find it 
the sun produces tides as well as the moon ; and 
the solar tides being at present much less than 
those originated by the moon, we have neglected 
them in the general outlines of the theory. The 
solar tides, however, must necessarily have an 
increasing significance. I do not mean that they 
will intrinsically increase, for there seems no 
reason to apprehend any growth in their actual 
amount; it is their relative importance to the 
lunar tides that is the augmenting quantity. As 
the final state is being approached, and as the 
velocity of the earth's rotation is approximating to 
the angular velocity with which the moon revolves 
around it, the ebbing and the flowing of the lunar 
tides must become of evanescent importance ; and 
this indeed for a double reason, partly on account 
of the moon's greatly augmented distance, and 


partly on account of the increasing length of the 
lunar day, and the extremely tardy movements of 
ebb and flow that the lunar tides will then have. 
Thus the lunar tides, so far as their dynamical 
importance is concerned, will ultimately approach 
to zero, while the solar tides retain their pristine 

We have therefore to examine the dynamical 
effects of solar tides on the earth and moon in 
the critical stage to which the present course of 
things tends. The earth will then rotate in a period 
of about fifty-seven of its present days ; and con- 
sidering that the length of the day, though so 
much greater than our present day, is still much 
less than the year, it follows that the solar tides 
must still continue so as to bring the earth's 
velocity of rotation to a point even lower than 
it has yet attained. In fact, if we could venture 
to project our glance sufficiently far into the 
future, it would seem that the earth must ulti- 
mately have its velocity checked by the sun-raised 
tides, until the day itself had become equal to 
the year. The dynamical considerations become, 
however, too complex for us to follow them, so 


that I shall be content with merely pointing out 
that the influence of the solar tides will prevent the 
earth and moon from eternally preserving the rela- 
tions of bending the same face towards each other; 
the earth's motion will, in fact, be so far checked, 
that the day will become longer than the month. 

Thus the doctrine of tidal evolution has con- 
ducted us to a prospect of a condition of things 
which will some time be reached, when the moon 
will have receded to a distance in which the month 
shall have become about fifty-seven days, and 
when the earth around which this moon revolves 
shall actually require a still longer period to ac- 
complish its rotation on its axis. Here is an odd 
condition for a planet with its satellite ; indeed, 
until a dozen years ago it would have been pro- 
nounced inconceivable that a moon should whirl 
round a planet so quickly that its journey was 
accomplished in less than one of the planet's own 
days. Arguments might be found to show that this 
was impossible, or at least unprecedented. There 
is our own moon, which now takes twenty-seven 
days to go round the earth ; there is Jupiter, with 
four moons, and the nearest of these to the 


primary goes round in forty-two and a half hours. 
No doubt this is a very rapid motion ; but all 
those matters are much more lively with Jupiter 
than they are here. The giant planet himself 
does not need ten hours for a single rotation, 
so that you see his nearest moon still takes be- 
tween two and three Jovian days to accomplish a 
single revolution. The example of Saturn might 
have been cited to show that the quickest revolu- 
tion that any satellite could perform must still 
require at least twice as long as the day in which 
the planet performed its rotation. Nor could the 
rotation of the planets around the sun afford the 
analogy of which we are in quest. For even 
Mercury, the nearest of all the planets to the sun 
of which the existence is certainly known, and 
therefore the most rapid in its revolution, requires 
eighty-eight days to get round once ; and in the 
meantime the sun has had time to accomplish 
between three and four rotations. Indeed, the 
planets and the older satellites would seem to have 
shown so great an improbability in the conclusion 
towards which tidal evolution points, that they 
would have contributed a serious obstacle to the 
general acceptance of that theory. 


But in 1877 an event took place so interesting 
in astronomical history, that we have to look back 
to the memorable discovery of Uranus in 1781 
before we can find a parallel to it in importance. 
Mars had always been looked upon as one of the 
moonless planets, though grounds were not want- 
ing for the surmise that probably moons to Mars 
really existed. It was under the influence of this 
belief that an attempt was made by Professor 
Asaph Hall at Washington to make a determined 
search, and see if Mars might not be attended by 
satellites large enough to be discoverable. The 
circumstances under which this memorable inquiry 
was undertaken were eminently favourable for its 
success. The orbit of Mars is one which possesses 
an exceptionally high eccentricity ; it consequently 
happens that the oppositions during which the 
planet is to be observed vary very greatly in the 
facilities they afford for a search like that con- 
templated by Professor Hall. It is obviously 
advantageous that the planet should be situated 
as near as possible to the earth, and in the op- 
position in 1877 the distance was almost at the 
lowest point it is capable of attaining ; but this 


was not the only point in which Professor Hall 
was favoured ; he had the use of a telescope of 
magnificent proportions and of consummate op- 
tical perfection. His observatory was also placed 
in Washington, so that he had the advantage of a 
pure sky and of a much lower latitude than any 
observatory in Great Britain is placed at. But the 
most conspicuous advantage of all was the practised 
skill of the astronomer himself, without which all 
these other advantages would have been but of 
little avail. Great success rewarded his well- 
designed efforts ; not alone was one satellite dis- 
covered which revolved around the planet in a 
period conformable with that of other similar cases, 
but a second little satellite was found, which accom- 
plished its revolution in a wholly unexpected and 
unprecedented manner. The day of Mars himself, 
that is, the period in which he can accomplish a 
rotation around his axis, very closely approximates 
to our own day, being in fact half an hour longer. 
This little satellite, the inner and more rapid of 
the pair, requires for a single revolution a period 
of only seven hours thirty-nine minutes, that is to 
say, the little body scampers more than three 


times round its primary before the primary itself 
has finished one of its leisurely rotations. Here 
was indeed a striking fact, a unique fact in our 
system, which riveted the attention of astronomers 
on this most beautiful discovery. 

You will now see the bearing which the move- 
ment of the inner satellite of Mars has on the 
doctrine of tidal evolution. As a legitimate con- 
sequence of that doctrine, we came to the con- 
clusion that our earth-moon system must ultimately 
attain a condition in which the day is longer than 
the month. But this conclusion stood unsupported 
by any analogous facts in the more anciently- 
known truths of astronomy. The movement of 
the satellite of Mars, however, affords the precise 
illustration we want ; and this fact, I think, adds 
an additional significance to the interest and the 
beauty of Professor Hall's discovery. 

It is of particular interest to investigate the 
possible connection which the phenomena of tidal 
evolution may have had in connection with the 
geological phenomena of the earth. We have 
already pointed out the greater closeness of the 
moon to us in times past. The tides raised by the 


moon on the earth must therefore have been 
greater in past ages than they are now, for of 
course the nearer the moon the bigger the tide. 
As soon as the earth and the moon had separated 
to a considerable distance we may say that the 
height of the tide will vary inversely as the cube 
of the moon's distance ; it will therefore happen, 
that when the moon was at half its present distance 
from us, his tide-producing capacity was not alone 
twice as much or four times as much, but even 
eight times as much as it is at present ; and a 
much greater rate of tidal rise and fall indi- 
cates, of course, a preponderance in every other 
manifestation of tidal activity. The tidal currents, 
for instance, must have been much greater in 
volume and in speed ; even now there are places 
in which the tidal currents flow at four or more 
miles per hour. We can imagine, therefore, the 
vehemence of the tidal currents which must have 
flowed in those diys when the moon was a much 
smaller distance from us. It is interesting to view 
these considerations in their possible bearings on 
geological phenomena. It is true that we have here 
many elements of uncertainty, but there is, how- 



ever, a certain general outline of facts which may 
be laid down, and which appears to be instructive, 
with reference to the past history of our earth. 

I have all through these lectures indicated a 
mighty system of chronology for the earth-moon 
system. It is true that we cannot give our chrono- 
logy any accurate expression in years. The various 
stages of this history are to be represented by the 
successive distances between the earth and the 
moon. Each successive epoch, for instance, may 
be marked by the number of thousands of miles 
which separate the moon from the earth. 

But we have another system of chronology de- 
rived from a wholly different system of ideas ; it too 
relates to periods of vast duration, and, like our 
great tidal periods, extends to times anterior to 
human history, or even to the duration of human 
life on this globe. The facts of geology open up 
to us a majestic chronology, the epochs of which 
are familiar to us by the succession of strata 
forming the crust of the earth, and by the succes- 
sion of living beings whose remains these strata 
have preserved. From the present or recent age 
our retrospect over geological chronology leads 


us to look through a vista embracing periods of 
time overwhelming in their duration, until at last 
our view becomes lost, and our imagination is 
baffled in the effort to comprehend the formation 
of those vast stratified rocks, a dozen miles or 
more in thickness, which seem to lie at the very 
base of the stratified system on the earth, and in 
which it would appear that the dawnings of life 
on this globe may be almost discerned. We have 
thus the two systems of chronology to compare 
one, the astronomical chronology measured by 
the successive stages in the gradual retreat of the 
moon ; the other, the geological chronology 
measured by the successive strata constituting 
the earth's crust. Never was a more noble problem 
proposed in the physical history of our earth than 
that which is implied in the attempt to correlate 
these two systems of chronology. What we would 
especially desire to know is the moon's distance 
which corresponds to each of the successive strata 
on the earth. How far off, for instance, was that 
moon which looked dowai on the coal forests in 
the time of their greatest luxuriance ? or what 

was the apparent size of the full moon at which 

K 2 


the ichthyosaurus could have peeped when he 
turned that wonderful eye of his to the sky on 
a fine evening ? But interesting as this great 
problem is, it lies, alas ! outside the possibility 
of exact solution. Indeed we shall not make any 
attempt which must necessarily be futile to cor- 
relate these chronologies ; all we can do is to 
state the one fact which is absolutely undeniable 
in the matter. 

Let us fix our attention on that specially inter- 
esting epoch at the dawn of geological time, when 
those mighty Laurentian rocks were deposited of 
which the thickness is so astounding, and let us 
consider what the distance of the moon must have 
been at this initial epoch of the earth's history. 
All we know for certain is, that the moon must 
have been nearer, but what proportion that distance 
bore to the present distance is necessarily quite 
uncertain. Some years ago I delivered a lecture 
at Birmingham, entitled "A Glimpse through the 
Corridors of Time," and in that lecture I threw 
out the suggestion that the moon at this primeval 
epoch may have only been at a fraction of its 
present distance from us, and that consequently 


terrific tides may in those days have ravaged the 
coast. There was a good deal of discussion on 
the subject, and while it was universally admitted 
that the tides must have been larger in palaeozoic 
times than they are at present, yet there was a 
considerable body of opinion to the effect that the 
tides even then may have been only about twice, 
or possibly not so much, greater than those tides 
we have at the present. What the actual fact 
may be we have no way of knowing ; but it is 
interesting to note that even the smallest accession 
to the tides would be a valuable factor in the per- 
formance of geological work. 

For let me recall to your minds a few of the 
fundamental phenomena of geology. Thosc-strati- 
fied rocks with which we are now concerned have 
been chiefly manufactured by deposition of sedi- 
ment in the ocean. Rivers, swollen, it may be, by 
floods, and turbid with a quantity of material held 
in suspension, discharge their waters into the sea. 
Granting time and quiet, this sediment falls to the 
bottom ; successive additions are made to its 
thickness during centuries and thousands of years, 
and thus beds are formed which in the course of 


ages consolidate into actual rock. In the formation 
of such beds the tides will play a part. Into the 
estuaries at the mouths of rivers the tides hurry in 
and hurry out, and especially during spring tides 
there are currents which flow with tremendous 
power ; then too, as the waves batter against the 
coast they gradually wear away and crumble do;vn 
the mightiest cliffs, and waft the sand and mud 
thus produced to augment that which has been 
brought down by the rivers. In this operation also 
the tides play a part of conspicuous importance, 
and where the ebb and flow is greatest it is obvious 
that an additional impetus will be given to the 
manufacture of stratified rocks. In fact, we may 
regard the waters of the globe as a mighty mill, 
incessantly occupied in grinding up materials 
for future strata. The main operating power of 
this mill is of course derived from the sun, for it 
is the sun which brings up the rains to nourish the 
rivers, it is the sun which raises the wind which 
lashes the waves against the shore. But there is 
an auxiliary power to keep the mill in motion, 
and that auxiliary power is afforded by the tides. 
If then we find that by any cause the efficiency 


of the tides is increased we shall find that the mill 
for the manufacture of strata obtains a correspond- 
ing accession to its capacity. Assuming the 
estimate of Professor Darwin, that the tide may 
have had twice as great a vertical range of ebb and 
flow within geological times as it has at present, 
we find a considerable addition to the efficiency 
of the ocean in the manufacture of the ancient 
stratified rocks. It must be remembered that the 
extent of the area through which the tides will 
submerge and lay bare the country, will often be 
increased more than twofold by a twofold increase 
of height. 

Suppose a hollow cone to be filled with water 
to a certain height, and let the quantity of water 
in it be measured ; now let the cone be filled 
until the water is at double the depth ; then the 
surfaces of the water in the two cases will be 
in the ratio of the circles, one of which has double 
the diameter of the other. The areas of the two 
surfaces are thus as four to one ; the volumes of the 
waters in the two cases will be in the proportion 
of two similar solids, the ratios of their dimensions 
being as two to one. Of course this means that 


the water in the one case would be eight times 
as much as in the other. This particular illustra- 
tion will not often apply exactly to tidal pheno- 
mena, but I may mention one place that I happen 
to know of, in the vicinity of Dublin, in which 
the effect of the rise and fall of the tide would 
be somewhat of this description. At Mala- 
hide there is a wide shallow estuary cut off from 
the sea by a railway embankment, and there is a 
viaduct in the embankment through which a great 
tidal current flows in and out alternately. At 
low tide there is but little water in this estuary, but 
at high tide it extends for miles inland. We may 
regard this inlet with sufficient approximation to 
the truth as half of a cone with a very large angle, 
the railway embankment of course forming the 
diameter ; hence it follows that if the tide was to 
be raised to double its height, so large an area of 
additional land would be submerged, and so vast an 
increase of water would be necessary for the pur- 
pose, that the flow under the railway bridge would 
have to be much more considerable than it is at 
present. In some degree the same phenomena will 
be repeated elsewhere around the coast. Simply 


multiplying the height of the tide by two would 
often mean that the border of land between high 
and low water would be increased more than two- 
fold, and that the volume of water alternately 
poured on the land and drawn off it would be 
increased in a still larger proportion. The velocity 
of all tidal currents would also be greater than 
at present, and as the power of a current of water 
for transporting solid material held in suspension 
increases rapidly with the velocity, so we may infer 
that the efficiency of tidal currents as a vehicle for 
the transport of comminuted rocks would be greatly 
increased. It is thus obvious that tides with a 
rise and fall double in vertical height of those 
which we know at present would add a large in- 
crease to their efficiency as geological agents. In- 
deed, even were the tides only half or one-third 
greater than those we know now, we might reason- 
ably expect that the manufacture of stratified rocks 
must have proceeded more rapidly than at present. 
The question then will assume this form. We 
know that the tides must have been greater .in 
Cambrian or Laurentian days than they are at 
present ; so that they were available as a means of 


assisting other agents in the stupendous oper- 
ations of strata manufacture which were then con- 
ducted. This certainly helps us to understand 
how these tremendous beds of strata, a dozen 
miles or more in solid thickness, were deposited. 
It seems imperative that for the accomplishment 
of a task so mighty, some agents more potent than 
those with which we are familiar should be re- 
quired. The doctrine of tidal evolution has shown 
us what those agents were. It only leaves us un- 
informed as to the degree in which their mighty 
capabilities were drawn upon. 

It is the property of science as it grows to find 
its branches more and more interwoven, and this 
seems especially true of the two greatest of all 
natural sciences geology and astronomy. With 
the beginnings of our earth as a globe in the shape 
in which we find it both these sciences are directly 
concerned. I have here touched upon another 
branch in which they illustrate and confirm each 

As the theory of tidal evolution has shed such a 
flood of light into the previously dark history of 
our earth-moon system, it becomes of interest to 


see whether the tidal phenomena may not have a 
wider scope ; whether they may not, for instance, 
have determined the formation of the planets by 
birth from the sun, just as the moon seems to have 
originated by birth from the earth. Our first 
presumption, that the cases are analogous, is not 
however justified when the facts are carefully 
inquired into. A principle which I have not 
hitherto discussed here assumes prominence, and 
therefore we shall devote our attention to it for a 
few minutes. 

Let us understand what we mean by the solar 
system. There is first the sun at the centre, which 
preponderates over all the other bodies so enor- 
mously, as shown in Fig. 4, in which the earth and 
the sun are placed side by side for comparison. 
There is then the retinue of planets, among the 
smaller of which our earth takes its place, a view 
of the comparative sizes of the planets being 
shown in Fig. 5. 

Not to embarrass ourselves with the perplexities 
of a problem so complicated as our solar system is 
in its entirety, we shall for the sake of clear reason- 
ing assume an ideal system, consisting of a sun and 


a large planet in fact, such as our own system 
would be if we could withdraw from it all other 
bodies, leaving the sun and Jupiter only remaining. 
We shall suppose, of course, that the sun is much 

Fig. 4. Comparative sizes of Earth and Sun. 

larger than the planet, in fact, it will be con- 
venient to keep in mind the relative masses of the 
sun and Jupiter, the weight of the planet being less 
than one-thousandth part of the sun. We know, of 
course, that both of those bodies are rotating upon 


their axes, and the one is revolving around the 
other ; and for simplicity we may further suppose 

Fig. 5. Comparative sizes of Planets. 

that the axes of rotation are perpendicular to the 
plane of revolution. In bodies so constituted tides 


will be manifested. Jupiter will raise tides in the 
sun, the sun will raise tides in Jupiter. If the 
rotation of each body be performed in a less period 
than that of the revolution (the case which alone 
concerns us), then the tides will immediately 
operate in their habitual manner as a brake for the 
checking of rotation. The tides raised by the sun 
on Jupiter will tend therefore to lengthen Jupiter's 
day ; the tides raised on the sun by Jupiter will 
tend to augment the sun's period of rotation. Both 
Jupiter and the sun will therefore lose some moment 
of momentum. We cannot, however, repeat too 
often the dynamical truth that the total moment 
of momentum must remain constant, therefore 
what is lost by the rotation must be made up 
in the revolution ; the orbit of Jupiter around the 
sun must accordingly be swelling. So far the 
reasoning appears similar to that which led to 
such startling consequences in regard to the moon. 
But now for the fundamental difference between 
the two cases. The moon, it will be remembered, 
always revolves with the same face towards the 
earth. The tides have ceased to operate there, and 
consequently the moon is not able to contribute 


any moment of momentum, to be applied to the 
enlargement of its distance from the earth ; all the 
moment of momentum necessary for this purpose 
is of course drawn from the single supply in the 
rotation of the earth on its axis. But in the case 
of the system consisting of the sun and Jupiter 
the circumstances are quite different Jupiter does 
not always bend the same face to the sun ; so 
far, indeed, is this from being true, that Jupiter 
is eminently remarkable for the rapidity of his 
rotation, and for the incessantly varying aspect in 
which he would be seen from the sun. Jupiter has 
therefore a store of available moment of momentum, 
as has also of course the sun. Thus in the sun 
and planet system we have in the rotations two 
available stores of moment of momentum on which 
the tides can make draughts for application to the 
enlargement of the revolution. The proportions 
in which these two available sources can be drawn 
upon for contributions is not left arbitrary. The 
laws of dynamics provide the shares in which each 
of the bodies is to contribute for the joint purpose 
of driving them further apart. 

Let us see if we camv t form an estimate by 


elementary considerations as to the division of the 
labour. The tides raised on Jupiter by the sun 
will be practically proportional to the sun's mass 
and to the radius of Jupiter. Owing to the enormous 
size of the sun, the efficiency of these tides and the 
moment of the friction-brake they produce will be 
far greater on the planet than will the converse 
operation of the planet be on the sun. Hence it 
follows that the efficiency of the tides in depriving 
Jupiter of moment of momentum will be greatly 
superior to the efficiency of the tides in depriving 
the sun of moment of momentum. Without 
following the matter into any close numerical calcu- 
lation, we may assert that for every one part the 
sun contributes to the common object, Jupiter will 
contribute at least a thousand parts ; and this 
inequality appears all the more striking, not to say 
unjust, when it is remembered that the sun is 
more abundantly provided with moment of mo- 
mentum than is Jupiter the sun has, in fact, about 
twenty thousand times as much. 

The case may be illustrated by supposing that a 
rich man and a poor man combine together to 
achieve some common purpose to which both are to 


contribute. The ethical notion that Dives shall 
contribute largely, according to his large means, and 
Lazarus according to his slender means, is quite 
antagonistic to the scale which dynamics has 
imposed. Dynamics declares that the rich man 
need only give a penny to every pound that has to 
be extorted from the poor man. Now this is pre- 
cisely the case with regard to the sun and Jupiter, 
and it involves a somewhat curious consequence. 
As long as Jupiter possesses available moment of 
momentum, we may be certain that no large con- 
tribution of moment of momentum has been 
obtained from the sun. For, returning to our 
illustration, if we find that Lazarus still has some- 
thing left in his pocket, we are of course assured 
that Dives cannot have expended much, because, as 
Lazarus had but little to begin with, and as Dives 
only puts in a penny for every pound that Lazarus 
spends, it is obvious that no large amount can have 
been devoted to the common object. Hence it 
follows that whatever transfer of moment of mo- 
mentum has taken place in the sun-Jupiter system 
has been almost entirely obtained at the expense 
of Jupiter. Now in the solar system at present, 



the orbital moment of momentum of Jupiter is 
nearly fifty thousand times as great as his present 
store of rotational moment of momentum. If, 
therefore, the departure of Jupiter from the sun 
had been the consequence of tidal evolution, it 
would follow that Jupiter must once have contained 
many thousands of times the moment of momentum 
that he has at present. This seems utterly incred- 
ible, for even were Jupiter dilated into an enor- 
mously large mass of vaporous matter, spinning 
round with the utmost conceivable speed, it is im- 
possible that he should ever have possessed enough 
moment of momentum. We are therefore forced 
to the conclusion that the tides alone do not provide 
sufficient explanation for the retreat of Jupiter 
from the sun. 

There is rather a subtle point in the consider- 
ations now brought forward, on which it will be 
necessary for us to ponder. In the illustration 
of Dives and Lazarus, the contributions of Lazarus 
of course ceased when his pockets were exhausted, 
but those of Dives will continue, and in the lapse 
of time may attain any amount within the utmost 
limits of Dives' resources. The essential point to 


notice is, that so long as Lazarus retains anything 
in his pocket, we know for certain that Dives has 
not given much ; if Lazarus, however, has his 
pocket absolutely empty, and if we do not know 
how long they may have been in that condition, we 
have no means of knowing how large a portion 
of wealth Dives may not have actually expended. 
The turning-point of the theory thus involves the 
fact that Jupiter still retains available moment 
of momentum in his rotation ; and this was our 
sole method of proving that the sun, which in 
this case was Dives, had never given much. But 
our argument must have taken an entirely 
different line had it so happened that Jupiter 
constantly turned the same face to the sun, and 
that therefore his pockets were entirely empty in 
so far as available moment of momentum is con- 
cerned. It would be apparently impossible for us 
to say to what extent the resources of the sun 
may not have been drawn upon ; we can, however, 
calculate whether in any case the sun could possibly 
have supplied enough moment of momentum to 
account for the recession of Jupiter. Speaking 
in round numbers, the revolutional moment of 

L 2 


momentum of Jupiter is about thirty times as great 
as the rotational moment of momentum at present 
possessed by the sun. I do not know that there is 
anything impossible in the supposition that the 
sun might, by an augmented volume and an aug- 
mented velocity of rotation, contain many times 
the moment of momentum that it has at this 
moment. It therefore follows that if it had hap- 
pened that Jupiter constantly bent the same face 
to the sun, there would apparently be nothing 
impossible in the fact that Jupiter had been born 
of the sun, just as the moon was born of the earth. 
These same considerations should also lead us 
to observe with still more special attention the 
development of the earth-moon system. Let us 
restate the matter of the earth and moon in the 
light which the argument with respect to Jupiter 
has given us. At present the rotational moment 
of momentum of the earth is about a fifth part 
of the revolutional moment of momentum of the 
moon. Owing to the fact that the moon keeps 
the same face to us, she has now no available 
moment of momentum, and all the moment of 
momentum required to account for her retreat has 


of late come from the rotation of the earth ; but 
suppose that the moon still had some liquid on 
its surface which could be agitated by tides, 
suppose further that it did not always bend the 
same face towards us, that it therefore had some 
available moment of momentum due to its rotation 
on which the tides could operate, then see how 
the argument would have been altered. The 
gradual increase of the moon's distance could be 
provided for by a transfer of moment of momentum 
from two sources, due of course to the rotational 
velocities of the two bodies. Here again the moon 
and the earth will contribute according to that 
dynamical but very iniquitous principle which 
regulated the appropriations from the purses of 
Dives and Lazarus. The moon must give not 
according to her abundance, but in the inverse 
ratio thereof because she has little she must give 
largely. Nor shall we make an erroneous estimate 
if we say that nine-tenths of the whole moment 
of momentum necessary for the enlargement of the 
orbit would have been abstracted from the moon; 
that means that the moon must once have had 
about five or six times as much moment of mo- 


mentum as the earth possesses at this moment. 
Considering the small size of the moon, this could 
only have arisen by terrific velocity of rotation, 
which it is inconceivable that its materials could 
ever have possessed. 

This presents the demonstration of tidal evolu- 
tion in a fresh light. If the moon now departed to 
any considerable extent from showing the constant 
face to the earth, it would seem that its retreat 
could not have been caused by tides. Some other 
agent for producing the present configuration would 
be necessary, just as we found that some other 
agent than the tides has been necessary in the case 
of Jupiter. 

But I must say a few words as to the attitude 
of this question with regard to the entire solar 
system. This system consists of the sun presiding 
at the centre, and of the planets and their satellites 
in revolution around their respective primaries, and 
each also animated by a rotation on its axis. I 
shall in so far depart from the actual configuration 
of the system as to transform it into an ideal 
system, whereof the masses, the dimensions, and 
the velocities shall all be preserved ; but that the 


several planes of revolution shall be all flattened 
into one plane, instead of being inclined at small 
angles as they are at present ; nor will it be un- 
reasonable for us at the same time to bring into 
parallelism all the axes of rotation, and to arrange 
that their common directions shall be perpendicular 
to the plane of their common orbits. For the 
purpose of our present research this ideal system 
may pass for the real system. 

In its original state, whatever that state may 
have been, a magnificent endowment was conferred 
upon the system. Perhaps I may, without deroga- 
tion from the dignity of my subject, speak of the 
endowment as partly personal and partly entailed. 
The system had of course different powers with re- 
gard to the disposal of the two portions ; the personal 
estate could be squandered. It consisted entirely of 
what we call energy ; and considering how frequently 
we use the expression conservation of energy, it 
may seem strange to say now that this portion 
of the endowment has been found capable of alien- 
ation, nay, further, that our system has been squan- 
dering it persistently from the first moment until 
now. Although the doctrine of the conservation 


of energy is, we have every reason to believe, a 
fundamental law affecting the whole universe, yet 
it would be wholly inaccurate to say that any 
particular system such as our solar system shall 
invariably preserve precisely the same quantity 
of energy without alteration. The circumstance that 
heat is a form of energy indeed negatives this 
supposition. For our system possesses energy of 
all the different kinds : there is energy due to the 
motions both of rotation and of revolution ; there 
is energy due to the fact that the mutually attract- 
ing bodies of our system are separated by distances 
of enormous magnitude ; and there is also energy 
in the form of heat ; and the laws of heat permit 
that this form of energy shall be radiated off into 
space, and thus disappear entirely, in so far as our 
system is concerned. On the other hand, there may 
no doubt be some small amount of energy accruing 
to our system from the other systems in space, 
which like ours are radiating forth energy. Any 
gain from this source, however, is necessarily so 
very small in comparison to the loss to which we 
have referred, that it is quite impossible that the 
one should balance the other. Though it is un- 


doubtedly true that the total quantity of energy in 
the universe is constant, yet the share of that 
energy belonging to any particular system such as 
ours declines steadily from age to age. 

I may indeed remark, that the question as to 
what becomes of all the radiant energy which the 
millions of suns in the universe are daily discharg- 
ing offers a problem apparently not easy to solve ; 
but we need not discuss the matter at present, we 
are only going to trace out the vicissitudes of our 
own system ; and whatever other changes that 
system may exhibit, the fact is certain that the 
total quantity of energy it contains is declining. 

Of the two endowments of energy and of moment 
of momentum originally conferred on our system 
the moment of momentum is the entailed estate. 
No matter how the bodies may move, no matter 
how their actions may interfere with one another, 
no matter how this body is pulled one way and the 
other body that way, the conservation of moment 
of momentum is not imperilled, nor, no matter 
what losses of heat may be experienced by radi- 
ation, could the store of moment of momentum be 
affected. The only conceivable way in which the 


quantity of moment of momentum in the solar 
system could be tampered with is by the inter- 
ference of some external attracting body. We 
know, however, that the stars are all situated at 
such enormous distances, that the influences they 
can exert in the perturbation of the solar system 
are <,bsolutely insensible ; they are beyond the 
reach of the most delicate astronomical measure- 
ments. Hence we see how the endowment of 
the system with moment of momentum has con- 
ferred upon that system a something which is 
absolutely inalienable. 

Before going" any further it would be necessary 
for me to explain more fully than I have hitherto 
done the true nature of the method of estimating 
moment of momentum. The moment of momentum 
consists of two parts : there is first that due to the 
revolution of the bodies around the sun ; there is 
secondly the rotation of these bodies on their axes. 
Let us first think simply of a single planet revolv- 
ing in a circular orbit around the sun. The 
momentum of that planet at any moment may 
be regarded as the product of its mass and its 
velocity ; then the moment of momentum of the 


planet in the case mentioned is found by multi- 
plying the momentum by the radius of the path 
pursued. In a more general case, where the planet 
does not revolve in a circle, but pursues an elliptic 
path, the moment of momentum is to be found by 
multiplying the planet's velocity and its mass into 
the perpendicular from the sun on the direction in 
which the planet is moving. 

These rules provide the methods for estimating 
all the moments of momentum, so far as the 
revolutions in our system are concerned. For the 
rotations somewhat more elaborate processes are 
required. Let us think of a sphere rotating round 
a fixed axis. Every particle of that sphere will of 
course describe a circle around the axis, and all 
these circles will lie in parallel planes. We may 
for our present purpose regard each atom of the 
body as a little planet revolving in a circular 
orbit, and therefore the moment of momentum of 
the entire sphere will be found by simply adding 
together the moments of momentum of all the 
different atoms of which the sphere is composed. 
To perform this addition the use of an elaborate 
mathematical method is required. I do not pro- 


pose to enter into the matter any further, except 
to say that the total moment of momentum is the 
product of two factors one the angular velocity 
with which the sphere is turning round, while the 
other involves the sphere's mass and dimensions. 

To illustrate the principles of the computation 
we shall take one or two examples. Suppose that 
two circles be drawn, one of which is double the 
diameter of the other. Let two planets be taken 
of equal mass, and one of these be put to revolve 
in one circle, and the other to revolve in the 
other circle, in such a way that the periods of 
both revolutions shall be equal. It is required to 
find the moments of momentum in the two cases. 
In the larger of the two circles it is plain that the 
planet must be moving twice as rapidly as in the 
smaller, therefore its momentum is twice as great ; 
and as the radius is also double, it follows that the 
moment of momentum in the large orbit will be 
four times that in the small orbit. We thus see 
that the moment of momentum increases in the 
proportion of the squares of the radii. If, however, 
the two planets were revolving about the same sun, 
one of these orbits being double the other, the 


periodic times could not be equal, for Kepler's 
law tells us that the square of the periodic time 
is proportional to the cube of the mean distance. 
Suppose, then, that the distance of the first 
planet is I, and that of the second planet is 2, 
the cubes of those numbers are I and 8, and there- 
fore the periodic times of the two bodies will be 
as i to the square root of 8. We can thus see that 
the velocity of the outer body must be less than 
that of the inner one, for while the length of the 
path is only double as large, the time taken to 
describe that path is the square root of eight times 
as great ; in fact, the velocity of the outer body 
will be only the square root of twice that of the 
inner one. As, however, its distance from the sun 
is twice as great, it follows that the moment of 
momentum of the outer body will be the square 
root of twice that of the inner body. We may 
state this result a little more generally as follows 
In comparing the moments of momentum of the 
several planets which revolve around the sun, that 
of each planet is proportional to the product of its 
mass with the square root of its distance from 
the sun. 


Let us now compare two spheres together, the 
diameter of one sphere being double that of the 
other, while the times of rotation of the two are 
identical. It can be shown by reasoning, into 
which I need not now enter, that the moment 
of momentum of the large sphere will be thirty- 
two times that of the small one. In general 
we may state that if a sphere of homogeneous 
material be rotating about an axis, its moment 
of momentum is to be expressed by the product 
of its angular velocity by the fifth power of its 

We can now take stock, as it were, of the con- 
stituents of moments of momentum in our system. 
We may omit the satellites for the present, while 
such unsubstantial bodies as comets and such small 
bodies as meteors need not concern us. The present 
investment of the moment of momentum of our 
system is to be found by multiplying the mass of 
each planet by the square root of its distance 
from the sun ; these products for all the several 
planets form the total revolutional moment of 
momentum. The remainder of the investment is 


in rotational moment of momentum, the collective 
amount of which is to be estimated by multiplying 
the angular velocity of each planet into its density, 
and the fifth power of its radius if the planet be 
regarded as homogeneous, or into such other 
power as may be necessary when the planet is 
not homogeneous. Indeed, as the denser parts 
of the planet necessarily lie in its interior, and have 
therefore neither the velocity nor the radius of the 
more superficial portions, it seems necessary to 
admit that the moments of momentum of the 
planets will be proportional to some lower power 
of the radius than the fifth. The total moment 
of momentum of the planets by rotation, when 
multiplied by a constant factor, and added to the 
revolutional moment of momentum, will remain 
absolutely constant. 

It may be interesting to note the present dis- 
position of this vast inheritance among the different 
bodies of our system. The biggest item of all is 
the moment of momentum of Jupiter, due to its 
revolution around the sun ; in fact, in this single 
investment nearly sixty per cent, of the total 
moment of momentum of the solar system is 


found. The next heaviest item is the moment of 
momentum of Saturn's revolution, which is twenty- 
four per cent. Then come the similar contri- 
butions of Uranus and Neptune, which are six 
and eight per cent, respectively. Only one more 
item is worth mentioning, as far as magnitude 
is concerned, and that is the nearly two per 
cent, that the sun contains in virtue of its rotation. 
In fact, all the other moments of momentum are 
comparatively insignificant in this method of view- 
ing the subject. Jupiter from his rotation has not 
the fifty thousandth part of his revolutional moment 
of momentum, while the earth's rotational share 
is not one ten thousandth part of that of Jupiter, 
and therefore is without importance in the general 
aspect of the system. The revolution of the earth 
contributes about one eight hundredth part of 
that of Jupiter. 

These facts as here stated will suffice for us to 
make a forecast of the utmost the tides can effect 
in the future transformation of our system. We 
have already explained that the general tendency 
of tidal friction is to augment revolutional moment 
of momentum at the expense of rotational. The 


total, however, of the rotational moment of mo- 
mentum of the system barely reaches two per cent, 
of the whole amount ; this is of course almost en- 
tirely contributed by the sun, for all the planets 
together have not a thousandth part of the sun's 
rotational moment. The utmost therefore that tidal 
evolution can effect in the system is to distribute the 
two per cent, in augmenting the revolutionary mo- 
ment of momentum. It does not seem that this can 
produce much appreciable derangement in the 
configuration of the system. No doubt if it were 
all applied to one of the smaller planets it would 
produce very considerable effect. Our earth, for 
instance, would have to be driven out to a distance 
many hundreds of times further than it is at 
present were the sun's disposable moment of mo- 
mentum ultimately to be transferred to the earth 
alone. On the other hand, Jupiter could absorb 
the whole of the sun's share by quite an insig- 
nificant enlargement of its present path. It does 
not seem likely that the distribution that must 
ultimately take place can much affect the present 
configuration of the system. 

We thus see that the tides do not appear to 



have exercised anything like the same influence 
in the affairs of our solar system generally which 
they have done in that very small part of the 
solar system which consists of the earth and moon. 
This is, as I have endeavoured to show in these 
lectures, the scene of supremely interesting tidal 
phenomena ; but how small it is in comparison 
with the whole magnitude of our system may be 
inferred from the following illustration. I repre- 
sent the whole moment of momentum of our 
system by 1,000,000,000, the bulk of which is 
composed of the revolutional moments of mo- 
mentum of the great planets, and the rotational 
moment of momentum of the sun. On this scale 
the rotational share which has fallen to our earth 
and moon does not even rise to the dignity of 
a single pound, it can only be represented by the 
very modest figure of igs. $d. This is divided 
into two parts the earth by its rotation accounts 
for 3-r. 4</., leaving i6s. id. as the equivalent of 
the revolution of the moon. The other inferior 
planets have still less to show than the earth. 
Venus can barely have more than 2s. 6d. ; even 
Mars' two satellites cannot bring his figure up 


beyond the slender value of \\d.\ while Mercury 
will be amply represented by the smallest coin 
known at her Majesty's mint. 

The same illustration will bring out the contrast 
between the Jovian system and our earth system. 
The rotational share of the former would be totally 
represented by a sum of nearly ; 12,000; of this, 
however, Jupiter's satellites only contribute about 
^"89, notwithstanding that there are four of them. 
Thus Jupiter's satellites have not one hundredth 
part of the moment of momentum which the 
rotation of Jupiter exhibits. How wide is the con- 
trast between this state of things and the earth- 
moon system, for the earth hardly contains in its 
rotation one-fifth of the moment of momentum 
that the moon has in its revolution ; in fact, the 
moon has gradually robbed the earth, which 
originally possessed igs. $d., and has carried it 
all off except 3^. ^d. 

And this process is still going on, so that ulti- 
mately the earth will be left very poor, though not 
absolutely penniless, at least if the retention of a 
halfpenny can be regarded as justifying that as- 
sertion. Saturn, revolving as it does with great 

M 2 


rapidity, and having a very large mass, possesses 
about 2700, while Uranus and Neptune taken 
together would figure for about the same amount. 

In conclusion, let us revert again to the two 
critical conditions of the earth-moon system. As to 
what happened before the first critical period, the 
tides tell us nothing, and every other line of reason- 
ing very little ; we can to some extent foresee 
what may happen after the second critical epoch is 
reached, at a time so remote that I do not venture 
even to express the number of ciphers which ought 
to follow the significant digit in the expression 
for the number of years. I mentioned, however, 
that at this time the sun tides will produce the 
effect of applying a still further brake to the rota- 
tion of the earth, so that ultimately the month will 
have become a shorter period than the day. It is 
therefore interesting for us to trace out the tidal 
history of a system in which the satellite revolves 
around the primary in less time than the primary 
takes to go round on its own axis such a system, 
in fact, as Mars would present at this moment were 
the outer satellite to be abstracted. The effect of 
the tides on the planet raised by its satellite 


would then be to accelerate its rotation ; for 
as the planet, so to speak, lags behind the tides, 
friction would now manifest itself by the con- 
tinuous endeavour to drag the primary round 
faster. The gain of speed, however, thus attained 
would involve the primary in performing more than 
its original share of the moment of momentum ; 
less moment of momentum would therefore remain 
to be done by the satellite, and the only way to 
accomplish this would be for the satellite to come 
inwards and revolve in a smaller orbit. 

We might indeed have inferred this from the 
considerations of energy alone, for whatever hap- 
pens in the deformation of the orbit, heat is pro- 
duced by the friction, and this heat is lost, and the 
total energy of the system must consequently 
decline. Now if it be a consequence of the tides 
that the velocity of the primary is accelerated, the 
energy corresponding to that velocity is also 
increased. Hence the primary has more energy 
than it had before ; this energy must have been ob- 
tained at the expense of the satellite ; the satellite 
must therefore draw inwards until it has yielded 
up enough of energy not alone to account for 


the increased energy of the primary, but also for 
the absolute loss of energy by which the whole 
operation is characterized. 

It therefore appears that in the excessively 
remote future the retreat of the moon will not 
only be checked, but that the moon may actually 
return to a point to be determined by the changes 
in the earth's rotation. It is, however, extremely 
difficult to follow up the study of a case where the 
problem of three bodies has become even more 
complicated than usual. 

The importance of tidal evolution in our solar 
system has also to be viewed in connection with 
the celebrated nebular hypothesis of the origin of 
the solar system. Of course it would be understood 
that tidal evolution is in no sense a rival doctrine 
to that of the nebular theory. The nebular origin 
of the sun and the planets sculptured out the main 
features of our system ; tidal evolution has merely 
come into play as a subsidiary agent, by which a 
detail here or a feature there has been chiselled into 
perfect form. In the nebular theory it is believed 
that the planets and the sun have all originated 
from the cooling and the contraction of a mighty 


heated mass of vapours. Of late years this theory, 
in its main outlines at all events, has strengthened 
its hold on the belief of those who try to interpret 
nature in the past by what we see in the present. 
The fact that our system at present contains some 
heat in other bodies as well as in the sun, and the 
fact that the laws of heat require continual loss by 
radiation, demonstrate that our system, if we look 
back far enough, and if the present laws have acted, 
must have had in part, at all events, an origin like 
that which the nebular theory would suppose. 

I feel that I have in the progress of these two 
lectures been only able to give the merest outline 
of the theory of tidal evolution in its application 
to the earth-moon system. Indeed I have been 
obliged, by the nature of the subject, to omit almost 
entirely any reference to a large body of the parts 
of the theory. I cannot bring myself to close these 
lectures without just alluding to this omission, and 
without giving expression to the fact, that I feel it 
is impossible for me to have rendered adequate 
justice to the strength of the argument on which 
we claim that tidal evolution is the most rational 
mode of accounting for the present condition in 


which we find the earth-moon system. Of course 
it will be understood that we have never contended 
that the tides offer the only conceivable theory as 
to the present condition of things. The argument 
lies in this wise. A certain body of facts are patent 
to our observation. The tides offer an explana- 
tion as to the origin of these facts. The tides are 
a vera causa, and in the absence of other suggested 
causes, the tidal theory holds the field. But much 
will depend on the volume and the significance of 
the group of associated facts of which the doctrine 
offers a solution. The facts that it has been in 
my power to discuss within the compass of dis- 
courses like the present, only give a very meagre 
and inadequate notion of the entire phenomena 
connected with the moon which the tides will 
explain. We have not unfrequently, for the sake 
of simplicity, spoken of the moon's orbit as circular, 
and we have not even alluded to the fact that 
the plane of that orbit is inclined to the ecliptic. 
A comprehensive theory of the moon's origin 
should render an account of the eccentricity of the 
moon's orbit ; it must also involve the obliquity of 
the ecliptic, the inclination of the moon's orbit, 


and the direction of the moon's axis. I have been 
perforce compelled to omit the discussion of these 
attributes of the earth-moon system, and in doing 
so I have inflicted what is really an injustice on 
the tidal theory. For it is the chief claim of the 
theory of tidal evolution, as expounded by Professor 
Darwin, that it links together all these various 
features of the earth-moon system. It affords a 
connected explanation, not only of the fact that the 
moon always turns the same face to the earth, but 
also of the eccentricity of the moon's path around 
the earth, and the still more difficult points about 
the inclinations of the various axes and orbits of 
the planets. It is the consideration of these points 
that forms the stronghold of the doctrine of tidal 
evolution. For when we find that a theory de- 
pending upon influences that undoubtedly exist, 
and are in ceaseless action around us, can at the 
same time bring into connection and offer a 
common explanation of a number of phenomena 
which would otherwise have no common bond of 
union, it is impossible to refuse to believe that such 
a theory does. actually correspond to nature. 

The greatest of mathematicians have ever found 


in astronomy problems which tax, and problems 
which greatly surpass, the utmost efforts of which 
they are capable. The usual way in which the 
powers of the mathematician have been awakened 
into action is by the effort to remove some glaring 
discrepancy between an imperfect theory and the 
facts of observation. The genius of a Laplace or 
a Lagrange was expended, and worthily expended, 
in efforts to show how one planet acted on another 
planet, and produced irregularities in its orbit ; 
the genius of an Adams and a Leverrier was 
nobly applied to explain the irregularities in 
the motion of Uranus, and to discover a cause 
of those irregularities in the unseen Neptune. 
In all these cases, and in many others which 
might be mentioned, the mathematician has 
been stimulated by the laudable anxiety to clear 
away some blemish from the theory of gravitation 
throughout the system. The blemish was seen to 
exist before its removal was suggested. In that 
application of mathematics with which we have 
been concerned in these lectures the call for the 
mathematician has been of quite a different kind. 
A certain familiar phenomenon on our sea-coasts 


has invited attention. The tidal ripples murmur a 
secret, but not for every ear. To interpret that 
secret fully, the hearer must be a mathema- 
tician. Even then the interpretation can only be 
won after the profoundest efforts of thought 
and attention, but at last the language has been 
made intelligible. The labour has been glori- 
ously rewarded, and an interesting chapter of 
our earth's history has for the first time been 

In the progress of these lectures I have sought 
to interest you in those profound investigations 
which the modern mathematician has made in his 
efforts to explore the secrets of nature. He has 
felt that the laws of motion, as we understand 
them, are bounded by no considerations of space, 
are limited by no duration of time, and he has 
commenced to speculate on the logical conse- 
quences of those laws when time of indefinite , 
duration is assumed to be at his disposal. From 
the very nature of the case, observations for con- 
firmation were impossible. Phenomena that re- 
quired millions of years for their development 


cannot be submitted to the instruments in our 
observatories. But this is perhaps one of the 
special reasons which make such investigations of 
peculiar interest, and entitle us to speak of the 
revelations of Time and Tide as a romance of 
modern science. 


ABERDEEN, tides at, 23 
Action and reaction, 69 
Adams, discoverer of Neptune, 186 
Admiralty Manual of Scientific In- 
quiry, 30, 31, 34 
Admiralty tide tables, 29 
Africa, Krakatoa dust over, 84 
Analysis, harmonic, of tides, 34 
Analysis of tide into its constitu- 
ents, 33 

Ancient tides on moon, 136 
Annus Magnus of solar system, 72 
Areas, conservation of, 61 
Arklow, tides at, 23 
Ascension, tides at, 24 
Astronomical chronology, 147 
Atlantic, Krakatoa dust over, 84 
Atlantic, tides in, 38 
Atmosphere, tides in, 40 
Avon, tides in, at Bristol, 39 

BALL-ROOM, illustration, 62 
Barometric records of Krakatoa air- 
wave, 82 

Batavia, Krakatoa heard at, 83 
Bath, hot waters at, 89 
Bay of Fundy, 39 
Beds of rock, how formed, 149 
Birmingham, lecture at, 148 
Birth of moon, 118 
Blast iron furnace, 87 
Blue sun produced by Krakatoa, 84 
Bodily tides of moon, 131 
Brake illustrating friction, 68 
Brickwork as a non-conductor, 87 
Bristol Channel, 39 
Bucket of water, oscillations in, 98 

CAMBRIAN rocks, 153 
Cannon-ball, energy of, 51 
Cardiff, tides at, 39 

Caspian Sea, tides in, 38 
Casting, cooling of, 79 
Celebes, Krakatoa heard at, 83 
Central America, 86 
Change and full, tides at, 22 
Chemical action in earth, 81 
Chcpstow, tides at, 39 
Chronology, the two systems o', 147 
Clifton, tides in Avon at, 39 
Clock, illustration of, 49 
Coal, ii 
Coincidence of moon's rotation and 

revolution, 134 
Combustion, heat of, 92 
Cones, volume of, 151 
Conservation of spin or areas, 64 
Constituent tides, 35 
Consumption of energy by tides, 48 
Cooling, laws of, 79 
Cooling of earth from primitive 

heat, 87 

Crane, brakes on, 69 
Craters of moon, 94 
Critical epoch in earth's history, 75 


Darwin, G. H., in Admiralty Manu- 
al 34 

Darwin, G. H. , on tidal evolution, 
97, 118, 122, 151, 185 

Day and month equal, 107 

Day at present increasing, 68 

Day of 1400 hours, 121 

Day of 3 or 4 hours, 75 

Decline of earth's heat, 90 

Diagram showing why moon re- 
cedes, 70 

DiegoGarcia, Krakatoa heard at, 83 

Difficulties of tidal evolution, 48 

Dives and Lazarus, 161 

Dust clouds, 84 



Dynamical principle, 65 
Dynamical stability, no 

EARTH and moon as rigid body, 107 

Earth a fly-wheel, 57 

Earth, fusion of, 12 

Earth, heat of, 80 

Earth in highly heated early state, 92 

Earth's crust, Lyell on, 12 

Earth's history, 12 

Earth-moon system originally, 107 

Earthquakes, 80 

Eccentricity of moon's path ex- 
plained, 184 

Ecliptic, obliquity of, explained, 184 

Economic aspects of tides, 45 

Egg on end, 108 

Elliptic orbit of moon, 113 

Endowment of moment of moment- 
um, 167 

Energy for tides, whence, 67 

Energy lost by tides, 60 

Energy of moon's position, 50 

Energy of motion, 51 

Energy of separation, 50 

Energy, sources of, 49 

Equality of day and month, 122 

Equilibrium , stable and unstable, 108 

Equinoxes, procession of, 72 

Eruption of volcanoes, 80 

Establishment, 30 

Estuary, tides in, 152 

Explosion of Krakatoa, 83 

Extinct season, moon, 129 

FIJI, 24 

Fishes, periods of fossil, 12 
Fitzgerald, Prof., on Hertz' experi- 
ments, 101 
Floating log, 109 
Fortnightly tide, 34 
Friction brake, 69 
Friction, tidal, 40, 47 
Full and change, tides during, 22 
Fundy, Bay of, tides in, 39 

GAUGE for tides, 31 
Geological chronology, 146 
Geology and tides, 144 

Geometric series, 92 

Geysers, 85 

Gibraltar, Straits of, tides in, 38 

Glimpse through the corridors of 

time, 148 

Glorious sunsets from Krakatoa, 85 
Greatest length of day, 121 
Greatest length of month, 122 
Greatest tides, 39 

Green moons from Krakatoa dust, 84 
Greenock, tides at, 22 
Grindstone, rupture of, 76 
Gunpowder, energy from, 51 

HALL, Prof. A., discovers satellites 

of Mars, 142 
Harmonic analyzer, 35 
Heated body, cooling of, 79 
Height of tide, how to measure, 29 
Helmholtz explains the constam 

face of the moon, 136 
Hertz, undulations of ether, 101 
High water, is it under the moon? 21 
High water, simple rules for, 26 
High water twice a day, 18 
Hot springs, 85 
Hot waters of Bath, 89 
Hour of high water, how found, 2 
Hovvth, 25 
Hull, tides at, 23 

Impulse, effect of timed, 101 
Incandescence of earth's interior, 85 
Incandescence of moon, 94 
Indian ocean, 84 . [j( 

Initial condition of earth and moon 
Instability, dynamical, no 
Interval, luni-tidal, 30 
Iron, machine for punching, 55 
Iron smelting, 87 

JAVA, 82 
Jupiter, 141, 156 

KEPLER'S Law, 53, 114 
Kerguelen Island, 24 
Kingstown, 21 
Krakatoa, 58 


I 9 T 


Language, origin of, 12 

Ln place, 186 

Lauren tian rocks, 148 

Law of tides, 15 

Lazarus and Dives, 161 

Leakage of heat, 86 

Length of day increasing, 68 

Leverrier discovers Neptune, 186 

London, tides at, 23, 27 

Lunar and solar tides compared, 15 

Lunar diurnal tide, 34 

Luni-tidal interval, 30 

Lyell, 12 

MACASSAR, explosion of Krakatoa 
heard at, 83 

Machine for analyzing tides, 35 

Machine for predicting tides, 35 

Machine for observing tides, 32 

Malahide estuary, 152 

Mars, moons of, 142 

Mass of moon found from tides, 16 

Max Muller, 12 

Maximum and minimum, 104 

Mediterranean, tides in, 38 

Meridian position of moon at high 
water, 26 

Mills, tidal, 43 

Mines, heat in, 86 

Moment of momentum, how esti- 
mated, 170 

Month equals day, 122 

Month of 1400 hours, 122 

Month with greatest number of 
days, 118 

Moon and tides connected, 13 

Moon, constant face of, 136 

Moon-energy, 54 

Moon, volcanic activity on, 94 

Moons of Mars, 142 

Motion, perpetual, 48 

NAUTICAL almanac, 26 
Neap tides, 16 
Needles, tides at, 23 
Niagara, utilization of, 47 
Noise from Krakatoa, 83 
Numerical data of tidal evolution, 127 

OBLIQUITY of ecliptic and tides, 184 

Observations of tides, 29 

Ocenn tides, 38 

Orbit of earth, changes in, 72 

Orbit of moon explained by tides, 184 

Origin of moon, 100 

Oscillations of water, 98 


Paradox of geometric series, 92 

Pendulum, motion of, 49 

Periodic phenomena, 71 

Periods of rotation and revolution 

of moon equal, 43 
Periods of tides, 34 
Perpetual motion, 48 
Planets and tides, 155 
Precession of equinoxes, 72 
Prediction of tides, 35 
Principal tides, 33 
Punching-engine, 55 
Purser, Professor, 61 

RAILWAY brakes, 68 

Reaction and action, 69 

Relative rotation, 104 

Relative tides on earth and moon, 131 

Reptiles, fossil, n 

Retreat of moon explained, 69 

Retrospect of moon's history, 77 

Rhine, mills on, 43 

Ring theory of moon's origin, 96 

Rivers, 149 

Rocks, formation of, 150 

Rodriguez, Krakatoa heard at, 83 

Rolling mills, 56 

Rotation of moon on its axis, 133 

Rupture of earth, 76 

Rupture of grindstone, 75 

ST. HELENA, tides at, 38 

Santa Cruz, 24 

Satellites of Jupiter, 135 

Saturn, 141 

Seasons, 72 

Seas, so called, on moon, 129 

Secondary ages, n 

Sligo, tides at, 23 

Small tides, 38 



Smelting of iron, 87 

Solar and lunar tides compared, 15 

Solar system, 155 

Solar tides, 14 [139 

Solar tides, ultimate importance of, 

Sounds from Krakatoa, 83 

Source of tidal energy, 67 

Spring tides, 16 

Stability, dynamical, no 

Stable equilibrium, 108 

Stages of special importance, 117 

Stalactites, 73 

Steam engine and tides compared, 45 

Strata, formation of, aided by tides, 


Sumatra, 82 
Sun, rotation of, 134 
Sunbeams stored, n 
Sunda, 82 
Sunsets, Krakatoa, 85 

TABLE of luni-tidal corrections, 30 

Telescope at Washington, 143 

Temperature of space, 91 

Tertiary ages, n 

Thames, tides in, 125 

Thomson, Sir W. , 36, 44 

Tidal currents used in making 

rocks, 153 

Tidal efficiency, how estimated, 14 
Tidal evolution, to whom due, 97 
Tide gauge, 31 
Tide mills, 44 
Tide predicting engine, 36 
Tides and geology, 144 
Tides at London bridge, 28 

Tides, change and full, 22 

Tides, commercial value of, 47 

Tides, due to moon, 13 

Tides, how to observe, 29 

Tides in Jupiter, 138 

Tides in moon, 128 

Tides, small, 38 

Tides, solar and lunar compared, 15 

Tides varying inversely as cube of 

distance, 15 
Time and tide, 12 
Timor, Krakatoa heard at, 83 
Tralee, tides at, 23 
Tyndall, 73 
Tynemouth, 23 

UNDULATIONS of a fluid globe, 99 
Undulations of air from Krakatoa, 82 
Unintermit'jng phenomena, 71 
Unstable equilibrium, 108 
Uranus, 142 

VAST tides, 39 
Volcanoes, 80 
Voyage of Krakatoa dust, 84 

WASHINGTON reflector, 142 
Water absent from moon, 129 
Water-wheels under London bridge, 


Waves of air from Krakatoa, 82 
Whewell on tides, 30 
Work done by tides, 42, 47 
Work of tides, 42 

YAKMOUTH, tides at, 23 


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